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    "title": "Blog",
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    "text": "Welcome to my Blog! I write about temporary interests, favorite open problems, neat arguments, random mathematical thoughts, things I want to remember, and stuff that is too little for a paper and too specific for Wikipedia. If somethings sparks your interest, feel free to contact me. Enjoy.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nConed frameworks and radial moves\n\n\n\n\n\n\nNov 19, 2024\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nBarycentric algebra\n\n\n\n\n\n\nNov 6, 2024\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe Mahler conjecture for coordinate symmetric bodies\n\n\n\n\n\n\nAug 8, 2024\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFrom finger moves to obstructions\n\n\n\n\n\n\nJul 20, 2024\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe double kissing problem\n\n\n\n\n\n\nJul 15, 2024\n\n\n\n\n\n\n\n\nNo matching items"
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    "title": "Teaching",
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    "text": "Lectures\n\n2024: Rigidity Theory for Frameworks and Polytopes (TU Berlin)\n2022: Polytope Theory (Warwick)\n2017: Computer Aided Geometric Design (TU Chemnitz)\n2016: Solid Modeling (TU Chemnitz)\n\nFor the full list of my teaching experience, see my CV.\n\n\nSupervising and Mentoring\n\nMaster theses\n\n2023: K. K. Gottwald, Approaching a Characterization of Generically Closed Orthogonal Matrix Groups\n2022: N. Nagel, Notes on the Union Closed Sets Conjecture\n\n\n\nThird-year projects\n\n2024: D. E. Türköz, Properties of Circulants and Related Graphs (Warwick)"
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    "title": "Research",
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    "text": "I work and publish on\n\npolyhedral combinatorics and geometry,\ngraph theory (spectral, algebraic, geometric and topological),\nrigidity theory, geometric constraint system and real algebraic geometry,\nlow-dimensional topology (knot theory, embeddings and embeddability of graphs and complexes, …)\nsymmetries of discrete and geometric objects,\nfinite group theory and real representation theory,\nalgebraic combinatorics (association schemes, distance regular graphs, codes …),\nconvex geometry,\nrandom structures and graph limits,\ndiscrete dynamical systems,\n…\n\nMy current main project is on the Wachspress Geometry of convex polytopes."
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    "title": "Lecture – Rigidity Theory for Frameworks and Polytopes",
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    "text": "When\n\n\nwinter semester 2024/25  every Tuesday, slot 10am – 12am (we meet 10:15)  October 15th, 2024 – February 11th, 2025\n\n\n\n\nWhere\\(\\;\\;\\)\n\n\nTU Berlin  MA 751 (Charlottenburg)\n\n\n\n\n!! Schedule changed !!\nSome lectures have been moved to a different day; the time remains 10:15 – 11:45am. The remaining lectures of 2024 are\n\nTuesday, November 26 (no lecture)\nFriday, November 29 (MA 541)\nTuesday, December 3\nFriday, December 6 (MA 541)\nTuesday, December 10 (no lecture)\nTuesday, December 17\n\nTo register for the course, visit the course website on ISIS (Information System for Instructors and Students). You can also find the course on Moses.\n\nDescription\nRigidity is a classical topic inspired from physics, engineering and architecture; but at the same time full of questions with intrinsically mathematical appeal and beauty.\nClassical rigidity theory studies frameworks which are graphs that are embedded in Euclidean space with straight-line edges. One should think of them physically as built from rigid metal rods that are connected at universal joints. The central question of rigidity theory is whether a given framework is flexible (it can be deformed in such a way that all its edges stay of the same length) or rigid (it cannot be deformed in this way). Questions of this nature have a long history in structural engineering, but are also surprisingly ubiquitous in pure mathematics. The question for rigidity turns out intricat, and so many tools have been developed to deal with it, either approximately or in special case. Today mathematicians study many different forms of rigidity (infinitesimal, global, universal, generic, minimal, …) and many different settings for rigidity (bar-joint frameworks, point-hyperplane frameworks, volume rigidity, but also rigidity of manifolds and polytopes). The subject is intrinsically geometric, using tools from and inspiring results in algebraic geometry, projective geometry, hyperbolic geometry and convex geometry.\nIn this lecture we start with an introduction to the classical rigidity theory of frameworks which still underlies all modern developments. Beginning from the core definitions we explore the first- and second-order theory. We will take a look at different forms and setting for rigidity, such global and generic rigidity, and also point-hyperplane frameworks and some other forms of rigidity. In the second part of the lecture we focus specifically on rigidity questions that arise in the geometric study of polytopes. Here we take a look at classical rigidity results by Cauchy, Dehn, Gluck, Alexandrov and Minkowski, but also modern developments and their connection to Wachspress Geometry.\n\n\nScope\nThe first half of the lecture (roughly week 1 – 8) will focus on the classical rigidity theory of frameworks. The second half (roughly week 9 – 16) will explore the rigidity theory of polytopes. My goal is to talk about the following topics:\nFrameworks\n\nfundamental concept: frameworks, tensegrities, flexes, rigidity (local, global, universal)\nfirst-order theory: infinitesimal flexes, stresses, rigidity matrix, projective invariance\nsecond-order theory: stress matrix, second-order rigidity, prestress stability, energy interpretation\ngeneric rigidity, Maxwell counting condition, rigidity matroid , Laman graphs\nglobal rigidity and generic global rigidity\nframeworks on the sphere, cylinder etc.\ncross-bracing theorems\npoint-hyperplane frameworks, volume rigidity\nsymmetry-forced rigidity, reduced counts, pure symmetry rigidity\n\nPolytopes\n\nrigidity theorems of Cauchy, Dehn and Gluck; Connelly’s flexible spheres, bellow theorem\npolyhedral frameworks, Tutte embeddings, reciprocal frameworks, the Maxwell-Cremona correspondece\nrigidity of triangulated surfaces\nrigidity of non-triangulated surfaces with coplanarity constraints\nconed polytope frameworks, the Wachspress-Izmestiev stress, stress-flex conjecture\nuniqueness theorem’s of Alexandrov, Minkowski, etc.\n\n\n\nPrerequisites\nThe prerequisits are minimal. I will assume a good grasp of linear algebra and some fundamental graph theory (connectivitiy, planarity, handshaking lemma, etc). For the second part of the lecture some background knowledge on polytopes (definition, combinatorics) can be advantageous, but we will also recall the essentials.\n\n\nLecture material\nI do not intend to provide lecture notes consistently. The material that was made available online can be found here.\n\n0 - Introduction (notes)\n1 - Frameworks & Rigidity (notes)\nI wrote a blog post on radial moves in coned frameworks."
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    "title": "Martin Winter",
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    "text": "Martin Winter, PhD   winter@math.tu-berlin.de\n\nI am a mathematician. I am Dirichlet fellow at TU Berlin and guest researcher at the Max-Planck-Institute for Mathematics in the Sciences in Leipzig. I am researching in discrete geometry, real algebraic geometry, geometic constraint systems, semi-definite optimization, spectral graph theory, algebraic combinatorics and low-dimensional topology.  Most recently I am studying the Wachspress Geometry of convex polytopes.\nI am principal investigator in the priority program “Combinatorial Synergies” and my project is “Wachspress Coordinates - a Bridge between Algebra, Geometry and Combinatorics”.\nBefore, I was postdoc at the University of Warwick (UK). I got my PhD from TU Chemnitz (Germany) in 2021.\nWhat’s new\n\nI am currently teaching the course “Rigidity Theory for Frameworks and Polytopes” at TU Berlin.\n\n\nWhat’s coming\n\n\n\n\nOn December 4 I am hosting Rainer Sinn at TU Berlin. He will speak in the Discrete Geometry seminar about “Positive Geometry in the plane”.\nOn December 5 I am at the workshop “LEAN meets MaRDI and OSCAR” at TU Berlin.\nI will be visiting the group “Differential Geometry and Geometric Structures” at TU Wien, January 7 – 10, 2025.\nI will be visiting the Combinatorial Group at the Czech Academy of Sciences, January 13 – 15, 2025. I will speak in their seminar on January 14.\nI will be visiting the group “Algorithmic Mathematics” at BTU Cottbus-Senftenberg on January 22 – 23, 2025. I will speak in their seminar on January 22.\nMarch 10 – 21, 2025 I will be visiting ICERM during the special semester on “Geometry of Materials, Packings and Rigid Frameworks”, and will take part in the workshop “Matroids, Rigidity, and Algebraic Statistics” (March 17 - 21, 2025)\n\nI am an active user of the StackExchange network. Here are links to my mathematical profiles:\n\nMathOverflow\nMath.StackExchange"
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    "title": "The double kissing problem",
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    "text": "Famously Isaac Newton and David Gregory disagreed over the maximal number of disjoint unit spheres that can simultaneously touch a central unit sphere. Newton claimed it to be 12, while Gregory thought that a 13-th sphere can fit. It took several centuries to show that Newton was right. Here is your chance to be part of a dispute just like that: make your guess for the double kissing problem.\n\nThe kissing number is the maximal number of disjoint (3-dimensional) unit spheres that can touch a common central unit sphere. Today we know the answer is 12. Some optimal solutions have noticable gaps between the outer spheres (see the right figure below) which made people wonder whether a 13-th sphere could fit.\n\nThe double kissing problem replaces the central sphere with two spheres:\n\nGiven two touching unit spheres, what is the maximal number of disjoint unit spheres that can be arranged so that each touches at least one of the central spheres?\n\nIt is not hard to come up with an arrangement of 18 spheres by taking a part of the fcc lattice (or A3 lattice):\n\nAs of writing, it seems that a 19-th sphere cannot made fit.\nHowever, in 2015 Moritz Firsching found that by shrinking the outer spheres by a tiny amount, to a radius of not less than 0.99, we actually can fit a 19-th sphere! Here is an animation of this arrangement based on coordinates provided by him on MathOverflow:\n\n\n\nWhile I am still rooting for double kissing number 18, this 19-sphere arrangement made me pessimistic regarding the existence of a slick argument for this.\nThe double kissing problem was initially brought to my attention by Florian Theil. In his paper “Face-Centered Cubic Crystallization of Atomistic Configurations” with Lisa Flatley they showed that the fcc lattice is (asymptotically) an energy minimizer for suitably chosen 2- and 3-point potentials. 2-point potentials (that is, the energy depends on the distance between pairs of particles) are familiar from the classical electrostatic and gravitation forces. In contrast, 3-point potentials (the energy depends on the distances in triples of particles) struck me as less natural (though Florian explained to me that they are not unphysical). In fact, Flatley and Theil conjecture that the 3-point potentials are actually not necessary for their result. They can prove this conditional on a new Conjecture 2.2, which reads (in my words):\n\nConjecture (Flatley, Theil, 2015)  Given a packing of unit sphere in which each sphere touches exactly 12 other spheres, then for any pair of touching spheres there are at least four spheres that touch both of them.\n\nIf the double kissing number were 18, then this would answers their conjecture affirmatively: let \\(B_0,B_1\\) be two touching spheres in an arrangement as in the conjecture, and let \\(N_i\\) be the set of spheres touching \\(B_i\\) (excluding \\(B_{1-i}\\)) then \\(|N_i|=11\\), \\(|N_0\\cup N_1|\\le 18\\), and\n\\[|N_0\\cap N_1| = |N_0|+|N_1|-|N_0\\cup N_1| \\ge 11+11-18 = 4.\\]\nTheir Conjecture 2.2 makes much stronger assumptions than the general double kissing problem and might therefore be more accessible. Note that in the animation of the 19-spheres configuration, there is not a single sphere that touches both central spheres. This is possible since the central spheres are not touched by 12 spheres each, but by 11 and 10 other spheres respectively (including the other central sphere)."
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    "title": "Short CV",
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    "text": "starting in 2025: principal investigator for DFG project (SPP 2458) at Max-Planck-Institute for Mathematics in the Sciences\nSince 2024: guest researcher at the Max-Planck-Institute for Mathematics in the Sciences\nSince 2024: Dirichlet fellow at TU Berlin\n2021 – 2024: Postdoc at the University of Warwick with Agelos Georgakopoulos\n2018 – 2021: Doctorate in Mathematics at TU Chemnitz in the group “Algorithmic and Discrete Mathematics” of Christoph Helmberg  Thesis: Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity  [thesis], [slides of defense talk]\n2016 – 2018: Research fellow at the working group of Computer Graphics at TU Chemnitz\n2013 – 2015: Master in Mathematics with Computer Science (minor in Physics) at TU Chemnitz\n2010 – 2013: Bachelor in Computational Science at TU Chemnitz (Germany)\n\nOr click here for my full CV (last updated: July 2024)."
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    "title": "Lecture – Polytope Theory",
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    "text": "I taught this TCC course on “Polytope Theory” starting from October 10th 2022 for 8 weeks. Due to the nature of TCC courses (being streamed to several universities) this was an online course taught via MS Teams.\n\nIntroduction\nPolytope Theory is the study of (convex) polytopes, the generalization of polygons (2D) and polyhedra (3D) to general dimension. Besides their geometric nature as convex sets, polytopes possess a rich combinatorial structure, making them exceptionally accessible by combinatorial techniques. The study of polytopes reaches from the antiquity (starting from the Platonic solids), over the 19th/20th century (understanding polytopes in 3D and initializing the study of polytopes in dimension \\(\\ge 3\\)), until today, where we understand that the richness of polytopal phenomena starts in dimension 4 and which led to findings such as universality. The subject has shown proximity to algebraic geometry, representation theory, analysis, optimization and many more.\n\n\nScope\nIn this course I aimed to give an overview of this very diverse subject and cover selected topics with focus towards research and open questions. I tried to cover the following, though not everything made it into the final course:\n\n\nrealization spaces and universality\n\n\nGale duality and enumeration of combinatorial types\n\n\nface numbers, Dehn-Sommerville equations and the Upper Bound Theorem\n\n\nreconstruction from the edge graph\n\n\nspectral theory of polytopes, expansion and Izmestiev’s Theorem\n\n\ngeometry and combinatorics of 3-dimensional polytopes\n\n\ninscribability and related geometric constraints\n\n\nsymmetry properties of polytopes\n\n\nzonotopes\n\n\nthe polytope algebra\n\n\n\n\nPrerequisites\nBesides an elementary geometric understanding, the prerequisites are minimal:\n\n\nlinear algebra\n\n\nelementary graph theory: basic definitions (sub-graph, vertex degree, bipartite graph), connectivity, handshaking lemma, planar graphs, etc\n\n\nbasic convex geometry: convex sets and cones, hyperplanes, some central theorems (Carathéodory’s theorem, hyperplane separation theorem) though we will give ample reminders for these\n\n\nbasic combinatorics: mostly some counting coefficients\n\n\nbasics of partially ordered sets, lattices, etc\n\n\nNot strictly necessary, but a background in any of the following will provide motivation and can help the understanding at some points: linear/convex optimization, algebraic topology, real representation theory of finite groups.\n\n\nLiterature\n\n\nB. Grünbaum, “Convex Polytopes” (a bit older, but with all the essentials)\n\n\nG. Ziegler, “Lectures on Polytopes” (modern, more focused on combinatorial aspects in the later chapters)\n\n\nA. Brøndsted, “An Introduction to Convex Polytopes”\n\n\nI. Pak, “Lectures on Discrete and Polyhedral Geometry” (not specific to polytopes, but contains many neat proofs; freely available)\n\n\n\n\nCourse notes\nThese course notes were created during each lecture. I am aware that there is the occasional typo, but overall they are correct.\n\n\n Lecture 1 (10/10/2022) Introduction (overview, motivation, applications), definition of polytope, V-polytopes, H-polytopes, Minkowski-Weyl theorem\n\n\n Lecture 2 (17/10/2022)Polar duals, faces and facets, face lattice, vertex figures\n\n\n Lecture 3 (24/10/2022)3-polytopes, edge-graphs, Steintz’ theorem, Balinski’s theorem, simple/simplicial polytopes, neighborly polytopes, cyclic polytopes, Gale’s evenness criterion, Kalai’s simple way to tell a simple polytope from its graph\n\n\n Lecture 4 (31/10/2022)Counting faces, Euler’s polyhedral formula + Euler Poincaré identity, Dehn-Sommerville equations, upper bound theorem, g-theorem\n\n\n Lecture 5 (07/11/2022)Polytopal complexes, shellability, line/linear shelling, Schlegel diagrams, abstract polytopal/simplicial complexes\n\n\n Lecture 6 (14/11/2022)Gale duality (linear/affine), spherical Gale diagrams, classifying small polytopes (d+1, d+2, d+3 vertices)\n\n\n Lecture 7 (21/11/2022) + GeoGebra files: addition, multiplication, squaring, golden ratio Realization spaces, centered realization space, (affine) Gale diagrams, point-line arrangements, Mnëv’s universality theorem, universality of 4-polytopes, non-rational polytopes\n\n\n\n Lecture 8 (28/11/2022)Selection of research directions in polytope theory"
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    "text": "On 2-complexes embeddable in 4-space, and the excluded minors of their underlying graphs  with Agelos Georgakopoulos\n\n\n\n[arXiv]\n\n\n\n\n\nEnergies on coned convex polytopes  with Robert Connelly, Steven J. Gortler, Louis Theran\n\n\n\n[preprint]\n\n\n\n\n\nThe stress-flex conjecture  with Robert Connelly, Steven J. Gortler, Louis Theran\n\n\n\n[arXiv]\n\n\n\n\n\nKalai’s \\(3^d\\) conjecture for unconditional and locally anti-blocking polytopes  with Raman Sanyal  accepted at Proceedings of the AMS\n\n\n\n[arXiv]\n\n\n\n\n\nThe clique graphs of the hexagonal lattice - an explicit construction and a short proof of divergence  Discrete Mathematics (2024) \n\n\n\n[arXiv]  [open access]\n\n\n\n\n\nCharacterizing Clique Convergence for Locally Cyclic Graphs of Minimum Degree \\(d\\ge 6\\)  with Anna M. Limbach  Discrete Mathematics (2024) \n\n\n\n[arXiv]  [open access]\n\n\n\n\n\nRigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints  International Mathematics Research Notices (2023)  This paper won the FSEM Post-Doctoral Research Prize 2024 from the University of Warwick\n\n\n\n[arXiv]  [open access]\n\n\n\n\n\n(Random) Trees of Intermediate Volume Growth  with George Kontogeorgiou  Extended abstract published at Eurocomb (2023)\n\n\n\n[arXiv]  [extended abstract]\n\n\n\n\n\nCapturing Polytopal Symmetries by Coloring the Edge-Graph  Discrete & Computational Geometry (2023) \n\n\n\n[arXiv]  [open access]\n\n\n\n\n\nEigenpolytopes, Spectral Polytopes and Edge-Transitivity\n\n\n\n[arXiv]\n\n\n\n\n\nThe Edge-Transitive Polytopes that are not Vertex-Transitive  with Frank Göring  Ars Mathematica Contemporanea (2022) \n\n\n\n[arXiv]  [link]\n\n\n\n\n\nSymmetric and Spectral Realizations of Highly Symmetric Graphs\n\n\n\n[arXiv]\n\n\n\n\n\nClassification of Vertex-Transitive Zonotopes  Discrete & Computational Geometry (2021) \n\n\n\n[arXiv]  [pdf]  [open access]\n\n\n\n\n\nGeometry and Topology of Symmetric Point Arrangements  Linear Algebra and its Applications (2021) \n\n\n\n[arXiv]  [link]\n\n\n\n\n\nVertex-Facet Assignments for Polytopes  with Thomas Jahn  Contributions to Algebra and Geometry (2020) \n\n\n\n[arXiv]  [open access]\n\n\n\n\nMy coauthors\n\nAgelos Georgakopoulos\nRobert Connelly\nSteven J. Gortler\nLouis Theran\nRaman Sanyal\nAnna M. Limbach\nGeorge Kontogeorgiou\nFrank Göring\nThomas Jahn"
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    "text": "2024\n\n“Rigidity and Reconstruction of Convex Polytopes – an Application of Wachspress Geometry” for my MATH+ Spotlight Talk (slides).\n“The Stress-Flex Conjecture – A riddle of rigidity in coned polytopes” at the Kick-Off Workshop of the SPP 2458 “Combinatorial Synergies” (slides).\n“Wachspress Coordinates - a bridge between algebra, geometry and combinatorics” at the Combinatorial Coworkspace in Kleinwalsertal (slides)\n“Rigidity of Polyhedral Spheres beyond Triangulations” at Landscapes of Rigidity Workshop during the RICAM Special Semester on Rigidity and Flexibility in Linz (slides)\n“Wachspress Objects and the Reconstruction of Convex Polytopes from Partial Data”, seminar talk during my visit at KTH Stockholm (slides)\n“Kalai’s 3d Conjecture for Coordinate Symmetric Polytopes” at the Workshop on Geometric and Algebraic Combinatorics in Santander (slides)\n\n\n\n2023\ncoming soon\n\n\n2022\ncoming soon\n\n\nOlder\nFor slides and further information on older talks you can visit my previous website at Warwick."
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    "title": "Wachspress Geometry",
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    "text": "Wachspress Geometry is a young subject in the intersection of geometry, combinatorics and algebra. It is at the core of my SPP 2458 project “Wachspress Coordinates – a bridge betwen Algebra, Geometry and Combinatorics”.\nWachspress Geometry is …\n\n… the study of the Wachspress objects, a somewhat mysterious family of constructions on convex polytopes. Those emerge in a wide range of seemingly unrelated contexts, and the goal is to explain and exploit these connections and to work towards a unifying explanation for the ubiquity of “Wachspress phenomena”.\n… among the best understood instances of Positive Geometry. The goal is to extend tools of Wachspress Geometry to more general positive geometries.\n\nPerhaps you have already come across some construction from Wachspress Geometry, whether in the shape of a canonical form, a stress in a framework, a model in algebraic statistics, cone volumes or volume differentials.\nBelow I will write an introduction to Wachspress Geometry, coming soon …\n\nIf you are interested, Rainer Sinn and I are organizing a workshop “Wachspress Geometry” in Leipzig on September 24th/25th 2024.\nmore coming soon …\n\nGeneralized barycentric coordinates\nPolytope skeleta as spectral embedding and the Izmestiev matrix\nCone volumes and variational definitions\nStresses in coned polytope frameworks + Maxwell-Cremona correspondence\nAdjoint polynomial and adjoint hypersurface + projective Geometry of Wachspress coordinated\nThe Wachspress map\nThe Wachspress variety\nThe Wachspress ideal\n\n\nGeneralized barycentric coordinates\nThere are several paths towards Wachspress Geometry. Here we follow an approach via generalized barycentric coordinates. For all that comes we fix a convex polytope \\(P\\subset\\Bbb R^d\\) with vertices \\(p_1,...,p_n\\in\\Bbb R^d\\).\nEach point \\(x\\in P\\) can be expressed as a convex combination of the polytope’s vertice, that is,\n\\[x = \\sum_i \\alpha_i p_i,\\quad \\text{where $\\alpha_i\\ge 0$ and $\\alpha_1+\\dots+\\alpha_n=1$}.\\]\nIf \\(P\\) is a simplex then there is a unique choice for the coefficients \\(\\alpha\\in\\Bbb R^n\\), which are known as barycentric coordinates of \\(x\\).  If \\(P\\) is not a simplex, there is more than one way for choosing the convex coefficients. For many applications it is however desirable to have a canonical choice across all points of \\(P\\), that is, a function \\(\\alpha:P\\to\\Delta_n\\) that satisfies linear precision:\n\\[\\sum_i \\alpha_i(x) p_i = x,\\quad\\text{for all $x\\in P$}.\\]\nHere \\(\\Delta_n:=\\{\\alpha\\in\\Bbb R^n_+\\mid \\alpha_1+\\cdots+\\alpha_n=1\\}\\) is the standard simplex or “space of convex coefficients”. Any such choice is called a system of generalized barycentric coordinates (or GBCs). Many such GBCs have been conceived for various application. One of them being the Wachspress coordinates.\n\n\nWachspress coordinates\nWachspress coordinates are a particular systems of generalized barcentric coordinates that exist for every convex polytope. Like many of the best constructions in mathematics, Wachspress coordinates have many equivalent definitions. In a sense, their true value for Wachspress Geometry lies in their many seemingly unrelated definitions that link them to a variety of subjects. We present some of them below.\n\n\nRational GBCs\nThe classical barycentric coordinates in simplices are linear functions in \\(x\\). Linear GBCs are not available for other polytopes, and so the next best choice, one might think, are polynomial GBCs. Eugene Wachspress however showed that there are polytopes for which no polynomial GBCs exist (in fact, most polytopes don’t have polynomial GBCs). Instead he constructed a systems of rational GBCs that we now call Wachspress coordinates. He defined them initially on polygons in 2D, which were later generalized by Warren to general polytopes in general dimension.\nAs rational GBCs Wachspress coordinates are of the form\n\\[\\alpha_i(x) = \\frac{\\mathrm p_i(x)}{\\mathrm q(x)},\\quad \\text{where $\\mathrm q(x)=\\sum_i\\mathrm p_i(x)$}.\\]\nwith polynomials \\(\\mathrm p_i,\\mathrm q\\in\\Bbb R[X_1,...,X_d]\\).\nIf you want to construct them explicitly, you will quickly find that they have to be of the following form:\n\\[\\mathrm p_i(x)=\\beta_i(x) \\prod_{F\\not\\ni i} H_F(x),\\]\nwhere the product runs over all facets \\(F\\) of \\(P\\) that are not incident to the vertex \\(p_i\\), \\(H_F\\) denotes the linear form that is zero on \\(\\operatorname{aff}(F)\\) and positive inside the polytope, and the \\(\\beta_i\\) are rational functions that might or might not be necessary to ensure normalization and linear precision. The reason for this form is that if \\(x\\) lies in a facet that does not contain \\(p_i\\), then this vertex cannot contribute to the convex combination for \\(x\\). This turns out to be the hard part: how to choose the \\(\\beta_i\\)? There is no easy answer, but one can compute them from facet volumes in the polar dual \\(P^\\circ\\). Most importantly however, the degree of \\(\\mathrm p_i\\) will eventually turn out precisely \\(\\text{\\#facets}-d\\).\n\nTheorem.  The Wachspress coordinates are the unique rational GBCs of lowest possible degree. In particular, \\(\\deg(\\mathrm p_i)=\\text{\\#facets}-d\\).\n\nWhere there are rational functions there are varieties and ideals: the image of \\(\\boldsymbol\\alpha:P\\to\\Delta_n\\) gives a variety (or a part of it) inside of the standard simplex. Its Zariski closure is known as the Wachspress variety, and the corresponding ideal is the Wachspress ideal. Both are objects of intrinsic algebro-geometric interest. Other objects are the adjoint polynomial and adjoint hyperpsurface of a polytope. …\n\n\nWPCs from cone volumes\nOne particularly geometrically pleasing definition of Wachspress coordiantes is via cone volumes. For this, let us assume that we want to compute the Wachspress coordinates of \\(x\\in \\operatorname{int}(P)\\),. We translate \\(P\\) so that \\(x=0\\).\n\\[P^\\circ := \\big\\{x\\in\\Bbb R^d\\mid \\langle x,p_i\\rangle \\text{ for all $i\\in\\{1,...,n\\}$}\\big\\}.\\]\nEach vertex \\(p_i\\) of \\(P\\) corresponds to a facet \\(F_i\\) of \\(P^\\circ\\). Let \\(C_i\\) be the cone over \\(F_i\\) with apex at the origin. The Wachspress coordiantes are then simply given by\n\\[\\alpha_i(x) := \\frac{\\operatorname{vol}(C_i)}{\\operatorname{vol}(P^\\circ)} = \\frac{\\operatorname{vol}(F_i)}{\\|p_i\\| \\operatorname{vol}(P^\\circ)}.\\]\nWe can also easily check that this satisfies linear precision:\n\\[\n\\sum_i \\alpha_i(x) p_i\n=\\sum_i \\frac{\\operatorname{vol}(F_i)}{\\|p_i\\| \\operatorname{vol}(P^\\circ)} p_i\n= \\frac1{\\operatorname{vol}(P^\\circ)} \\sum_i n_i \\operatorname{vol}(F_i)\n= 0,\n\\]\nwhere \\(n_i:=p_i/\\|p_i\\|\\) is the normal vector of the facet \\(F_i\\), and the last equality is therefore simply Minkowski’s balancing condition for polytopes.\n\n\nThe Lovász-Izmestiev matrix\nI personally encountered Wachspress coordinates first by studying the spectral graph theory of polytope skeleta. I found that Wachspress coordinates are merely a shadow of a higher rank objects.\nLet \\(G=(V,E)\\) be a graph. Spectral graph theory studies the spectral properties of matrices associated to \\(G\\), such as the adjacency matrix \\(A\\) or the Laplace matrix \\(L\\). It can also be used to construct embeddings of \\(G\\) into Eucliden space that reflect some of the combinatorial properties of \\(G\\) geometrically.\nFor example, fix an eigenvalue \\(\\theta\\in\\operatorname{Spec}(A)\\) and a basis \\(u_1,...,u_d\\in\\Bbb R^n\\) of the associated eigenspace \\(\\operatorname{Eig}_G(\\theta)\\). Let \\(\\Phi\\in\\Bbb R^{n\\times d}\\) be the matrix that has the \\(u_i\\) as its columns. Its rows correspond the the vertices of the graph, and so by reading out the matrix row-wise, we obtain an embedding of \\(G\\) into \\(d\\)-dimensional Euclidean space.\n\\[...\\]\nThis is called a spectral embedding of \\(G\\) w.r.t. \\(\\theta\\). Being a spectral embedding is a very special property, and most embedding are not spectral. The surprising result due to Lovász (\\(d=3\\)) and Izemstiev (\\(d> 3\\)) was that the skeleton of a polytope \\(P\\) is always a spectral embedding of the edge graph \\(G_P\\), assuming a suitable weighting of the vertices and edges.\n\nTheorem. (Lovász, Izmestiev)  There is a matrix \\(M\\in\\Bbb R^{n\\times n}\\) so that …\n\nasdasd\nadasda\nadasd\n\n\n\nThe connection to Wachspress Geometry is the following:\n\\[\\alpha_i(x)=\\sum_j M_{ij}(x),\\quad\\text{for all $i\\in \\{1,...,n\\}$}.\\]\n\n\nVolume variations\n\\[\\mathop{\\mathrm{vol}}(P^\\circ(\\boldsymbol c)) = \\mathop{\\mathrm{vol}}(P^\\circ) + \\langle\\tilde{\\boldsymbol\\alpha},\\boldsymbol c-\\boldsymbol 1\\rangle + \\tfrac12 (\\boldsymbol c-\\boldsymbol 1)^\\top \\tilde M (\\boldsymbol c - \\boldsymbol 1) + \\cdots \\]\n\n\nStresses in coned polytope frameworks\n\nStress-flex conjecture"
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    "title": "Tuning the geometric join",
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    "text": "The join is a well-known and rather general operation – it works on topological spaces, simplicial complexes, subsets of \\(\\Bbb R^n\\), polytopes and many others. I want to collect here some facts about joins that are perhaps not so well-known. This is also in preparation for an upcoming blog post on the non-symmetric Mahler conjecture. \\(\\DeclareMathOperator{\\vol}{vol}\\)\n\n\nThe join is an operation defined on objects as general as topological spaces. Given topological spaces \\(X\\), \\(Y\\), their join \\(X\\star Y\\) is defined as \\(X\\times [-1,1]\\times Y\\) quotiented by \\(x\\times\\{-1\\}\\times Y\\) for all \\(x\\in X\\) and \\(X\\times \\{1\\}\\times y\\) for all \\(y\\in Y\\). Already in this form it comes with some nice properties, e.g. \\(\\Bbb S^{d_1}\\star \\Bbb S^{d_2} \\simeq\\Bbb S^{d_1+d_2+1}\\).\nIf \\(X_i\\subset\\Bbb R^{d_i}\\) then \\(X_1\\star X_2\\) can be realized as a subset of \\(\\Bbb R^{d_1}\\times\\Bbb R\\times \\Bbb R^{d_2}\\cong\\Bbb R^{d_1+d_2+1}\\) as follows: embedd \\(X_1,X_2\\) into \\(\\Bbb R^{d_1+d_2+1}\\) so that their affine spans are disjoint. Then embed \\((x,t,y)\\) as \\((1-t)x+ty\\).\nIf the \\(X_i\\) are convex bodies, then this yields precisely the convex hull of \\(X_1^\\star\\cup X_2^\\star\\),in partiucular, is itself realized as a convex set.\nIf the \\(P_i\\) are polytopes, then their join \\(P_1\\star P_2\\) is a polytope whose combinatorics depends only on the combinatorics of the \\(P_i\\). Therefore, the join can also be defined as an operation on (face) lattices \\(\\mathcal P_i\\). In this form, some interesting properties are:\n\nthe join of simplices yields a simplex: \\(\\Delta_{d_1}\\star \\Delta_{d_2}\\simeq\\Delta_{d_1+d_2+1}\\) (combinatorially clear, just count the number of vertices).\nthe join commutes with duality: \\((\\mathcal P_1\\star \\mathcal P_2)^*\\simeq \\mathcal P_1^*\\star \\mathcal P_2^*\\), where \\(\\mathcal P^*\\) denotes combinatorial dual to the lattice \\(\\mathcal P\\) (i.e. turning it up side down).\n\nIt is now a good exercise to see whether these properties can be made work in the geometric setting as well. For example, if \\(0\\in P_i\\), can we have \\((P_1\\star P_2)^\\circ=P_1^\\circ\\star P_2^\\circ\\)? Can we have that the join of regular simplices is regular? How to compute the volume of \\(P_1\\star P_2\\)? These particular questions are inspired by their application to the non-symmetric Mahler conjecture in this blog article.\n\nIf the spaces are subsets \\(X_1\\subset\\Bbb R^{d_1}\\) and \\(X_2\\subset\\Bbb R^{d_2}\\), then the join \\(X_1\\star X_2\\) can be realized as a subset of \\(\\Bbb R^{d_1}\\times\\Bbb R\\times\\Bbb R^{d_2}\\cong\\Bbb R^{D}\\) (where \\(D=d_1+d_2+1\\)): embed \\(X_1\\) and \\(X_2\\) into \\(\\Bbb R^D\\) so that their affine spans are disjoint, and then mapping \\((a,t,b)\\) to \\((1-t)a+tb\\).\nLet’s be more precise. We fix disjoint subspaces \\(H_1,H_2\\subset \\Bbb R^{D}\\) of dimension \\(d_1,d_2\\) respectively, as well as isomorphisms \\(\\phi_i:\\Bbb R^{d_i}\\xrightarrow{\\sim} H_i\\).\n\nThe faces of \\(P_1\\star P_2\\) are of the form \\(\\sigma_1\\star \\sigma_2\\), where \\(\\sigma_i\\in\\mathcal F(P_i)\\). Hence, the f-vector of the join is\n\\[f_k(P_1\\star P_2)= \\sum_{i+j=k} f_i(P_1)f_j(P_2).\\]\nIn particular, the faces can be indexed by pairs \\((\\sigma_1,\\sigma_2)\\) and the face lattice structure can be recovered via\n\\[(\\sigma_1,\\sigma_2)\\preceq (\\tau_1,\\tau_2) \\;:\\Longleftarrow\\; \\sigma_1\\preceq \\tau_1\\,\\land\\,\\sigma_2\\preceq\\tau_2.\\]\nIn fact, the face lattice of the join is merely the product of the individual face lattices (in the sense of the usual product order). From this it is easy to see that\n\\[\\big(\\mathcal F(P_1)\\star \\mathcal F(P_2)\\big)^{\\mathrm{op}} \\simeq \\mathcal F(P_1)^{\\mathrm{op}}\\star\\mathcal F(P_2)^{\\mathrm{op}}.\\]\n\nLet’s fix parameter \\(x_1,x_2,c_1,c_2\\in\\Bbb R\\). For convex sets \\(B_1\\subset\\Bbb R^{d_1}\\) and \\(B_2\\subset\\Bbb R^{d_2}\\) we write \\(B_1 \\star B_2 \\subset \\Bbb R^{d_1}\\times\\Bbb R^{d_2}\\times\\Bbb R\\cong \\Bbb R^{d_1+d_2+1}\\) for the result of the following construction:\n\nconsider \\(\\Bbb R^{d_1+d_2+1}=\\operatorname{span}\\{e_0;e_1^1,...,e_{d_1}^1;e_1^2,...,e_{d_2}^2\\}\\) and define the following affine subspaces:  \\[\nH_i=x_i e_0 + \\operatorname{span}\\{e_{1}^i,...,e_{d_2}^i\\}.\n\\]\nembed \\(c_i B_i\\) into \\(H_i\\) with the origin mapped to \\(x_i e_0\\).\nthe convex hull of both embedded sets is the result of the operation.\n\nThe other direction requires some preparation. Let \\(B_1\\star B_2 \\subset\\Bbb R^{2d+1}\\) denote the join of convex bodies \\(B_1,B_2\\subset\\Bbb R^d\\). For our purpose we need that the join is constructed geometrically as follows: first we embed the \\(B_i\\) into two specific \\(d\\)-dimensional affine subspaces of \\(\\Bbb R^{2d+1}\\):\n\\[\n\\begin{align}\nH_1 &= -e_0 + \\operatorname{span}\\{e_1,...,e_d\\},\n\\\\\nH_2 &= \\phantom+ e_0 + \\operatorname{span}\\{e_{d+1},...,e_{2d}\\},\n\\end{align}\n\\]\nso that the origin in \\(B_i\\) is mapped to \\(\\pm e_0\\) respectively. The join \\(B_1\\star B_2\\) is then the convex hull of the embedded sets.\nWe make three obsevations.\n(1) Volume\nThe volume of \\(B_1\\star B_2\\) can be computed as as follows: the section of \\(B_1\\star B_2\\) with the hyperplane \\(e_0^\\bot + se_0, s\\in[-1,1]\\) is of the form \\(t B_1\\times (1-t) B_2\\) with \\(t=\\frac12(s+1)\\in[0,1]\\). Thus\n\\[\n\\begin{align}\n\\vol(B_1\\star B_2)\n    &= \\phantom 2\\!\\int_{-1}^1 \\!\\!\\vol\\big((B_1\\star B_2) \\cap (e_0^\\bot+ s e_0)\\big) \\,\\mathrm ds\n\\\\ &= 2\\! \\int_0^1 \\!\\!\\vol(t B_1\\times (1-t) B_2) \\,\\mathrm dt\n\\\\ &= \\vol(B_1)\\cdot \\vol(B_2) \\cdot \\underbrace{2\\!\\int_0^1 \\!\\!t(1-t) \\,\\mathrm dt}_{=: c}.\n    = c\\cdot \\vol(B_1)\\cdot\\vol(B_2).\n\\end{align}\n\\]\n(2) Polarity\nThe specific gemetric realization of the join chosen here commutes with polarity: \\[(B_1\\star B_2)^\\circ=B_1^\\circ\\star B_2^\\circ.\\] In particular, \\(B\\star B^\\circ\\) is self-polar!\n(2) Simplices\nSimplices are closed under the join operation. If \\(\\Delta_d\\) denotes a \\(d\\)-dimensional simplex, then\n\\[\\Delta_d\\star\\Delta_e \\simeq \\Delta_{d+e+1}.\\]\n\n\\(s=(1-t)a+tb = a + (b-a)t\\).\n\\[\n\\begin{align}\n\\vol(A\\star B)\n    &= \\int_a^b \\!\\vol\\!\\big((A\\star B)\\cap H_s\\big) \\,\\mathrm ds\n  \\\\&= (b-a)\\cdot \\int_0^1 \\!\\vol\\!\\big((1-t)\\alpha A+t \\beta B\\big) \\,\\mathrm dt\n  \\rlap{\\quad \\text{\\small\\color{lightgray} (using $\\tfrac{\\mathrm ds}{\\mathrm dt}=b-a$)}}\n  \\\\&= (b-a) \\cdot\\alpha^{d_A}\\beta^{d_B} \\cdot\\vol(A)\\cdot\\vol(B)\\cdot \\int_0^1 (1-t)^{d_A}t^{d_B} \\,\\mathrm dt\n  \\\\&= \\underbrace{\\alpha^{d_A}\\beta^{d_B}\\cdot(b-a)\\cdot\\frac{(d_A+1)!(d_B+1)!}{(d_A+d_B+1)!}}_{=:\\,K(a,b;\\alpha,\\beta;d_A,d_B)} \\cdot\\vol(A)\\cdot\\vol(B).\n\\end{align}\n\\]"
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    "title": "The stress-flex conjecture",
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    "text": "While studying\nLet \\(P\\) be a polytope with edge graph \\(G\\) and an interior point \\(x\\in\\operatorname{int}(P)\\). The corresponding coned polytope framework \\((G,\\boldsymbol{p})\\) has as its bars the edges of \\(P\\) together with additional bars connecting the polytope’s vertices to \\(x\\).\nIn general, showing that a given framework is rigid is a complicated matter. If one is lucky, the framework is first- or second-order rigid, which is easier to verify.\nUsing tools from Wachspress Geometry …\n\nTheorem.   Coned polytope frameworks are rigid. \n\n…\n\nThe stress-flex conjecture.   Given a coned polytope framework \\((G,\\boldsymbol{p})\\). For any first-order flex \\(\\dot{\\boldsymbol{p}}\\) and any stress \\(\\boldsymbol{\\omega}\\) holds  \\[\\sum_{i\\in V} \\omega_i\\kern1pt \\dot p_i = 0.\\]\n\nWe checked the conjecture in a wide range of setting:\n\nfor cone points outside the polytope\nfor non-convex polytopes\nfor closed surfaces of higher genus\nfor higher-dimensional polytopes\n\nWe note that the conjecture does not hold in all cases:\n\nwe can scale a vertex radially ourward. This preserves stresses and flexes, but not the stress-flex orthogonality condition.\nspectral embeddings of sparse graphs have both stresses and flexes. But they do not necessarily satisfy the stress-flex conjecture.\n\nThe intial motivation for the stress-flex conjecture, and a consequence in case it is solved affirmatively, is the second-order rigidity of coned polytope frameworks.\n\nA convex geometry conjecture\nLet \\(P(t),t\\in\\Bbb R\\) be a “differentiably parametrized” family of polytopes, i.e., \\(P(t)\\) is given by its facet hyperplanes \\(H_i(t)\\) which vary differentiably. Note that the polytopes are not necessarily of the same combinatorial type, but have the same number of facets (at least locally).\nIf \\(n_i\\) and \\(V_i\\) are the normal vector and volume of the \\(i\\)-th facet respectively, then Minkowski’s balancing condition states \\(\\sum_i n_i V_i=0\\). Differentiating this yields\n\\[\\sum_i \\dot n_i V_i + \\sum_i n_i \\dot V_i=0.\\]\nI conjecture that under certain conditions the terms vanish individually:\n\nConjecture   If the dihedral angles \\(\\theta_{ij} = \\arccos\\langle n_i,n_j\\rangle\\) do not change in first order, that is \\(\\dot\\theta_{ij}=0\\) for all edges \\(ij\\), then  \\[\\sum_i \\dot n_i V_i = \\sum_i n_i\\dot V_i =0.\\]\n\nSome notes:\n\nThe easiest way to preserve dihedral angles is by not changing normal vector direction, that is, to stay within a class of normally equivalent polytopes. In this case \\(\\dot n_i=0\\) and the conjecture holds trivially.\nIn the case where the combinatorics does not change we are in the setting of the infinitesimal Stoker’s conjecture which was recently solved. It states that from \\(\\dot\\theta_{ij}=0\\) follows \\(\\dot n_i=0\\) (after cancelling rotations). So the conjecture holds here as well.\nThe stres-flex conjecture for Wachspress coordinates is a special case of the above conjecture if viewed from the dual: if in \\(P\\) radii and edge lengths don’t change, then in the dual \\(P^\\circ\\) dihedral angles don’t change. The vanishing of \\(\\sum_i \\dot \\dot n_i V_i=\\sum_i (n_i\\|p_i\\|)(V_i/\\|p_i\\|)=\\sum_i \\dot p_i \\alpha_i=0\\) is percisely the claim of the stress-flex conjecture."
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    "title": "The Mahler conjecture for coordinate symmetric bodies",
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    "text": "The Mahler conjecture asserts that among all centrally symmetric convex bodies \\(B\\) the cube minimizes the volume product \\(\\operatorname{vol}(B)\\operatorname{vol}(B^\\circ)\\). While famously open, several proofs have been discovered for the special case of coordinate symmetric bodies. I want to share here a surprisingly short and elegant proof by Mathieu Meyer that seems otherwise unavailable in the English litrature. \\(\\DeclareMathOperator{\\vol}{vol}\\)\n\n\nLet \\(B\\subset\\Bbb R^n\\) be a centrally symmetric convex body, where we shall assume \\(-B=B\\). Kurt Mahler defined the following linearly invariant notion of volume now known as Mahler volume:\n\\[\nM(B):=\\operatorname{vol}(B)\\operatorname{vol}(B^\\circ).\n\\]\nHere \\(B^\\circ:=\\{y\\in\\Bbb R^n\\mid \\langle x,y\\rangle\\le 1\\text{ for all }x\\in B\\}\\) denotes the polar dual of \\(B\\). It has long been known that the Mahler volume attains a maximum for the sphere. There are also clear ideas for what should minimize the Mahler volume: the cube, and more generally, each of the Hanner polytopes. But a proof remains elusive. This is the infamous Mahler conjecture, which remains open to this day:\n\nMahler conjecture.  For each centrally symmetric convex body \\(B\\subset\\Bbb R^n\\) holds \\[M(B)\\ge M(\\text{$n$-cube}) = \\smash{\\frac{4^n}{n!}}.\\]\n\nBelow I share a very elegant and somewhat mysterious proof for the special case of coordinate symmetric bodies that, I believe, is not available in the English literature elsewhere. For some general background on the Mahler conjecture as well as its status, check out the Wikipedia article and Terry Tao’s blog entry on the topic.\n\nCoordinate symmetric bodies and convex corners\nA convex body \\(B\\subset\\Bbb R^n\\) is coordinate symmetric if it is invariant under reflection on each coordinate hyperplane \\(H_i:=\\{x_i=0\\}\\). A number of properties make coordinate symmetric bodies a natural special case for studying the Mahler conjecture: they are centrally symmetric, closed under polarity, and they contain all Hanner polytopes.  \n\nIn the literature these bodies are often called “unconditional bodies”, a term going back to “unconditional convergence” motivated from Banach space theory. For geometric (that is, finite-dimensional) discussions I prefer to go with the more visually inspired term “coordinate symmetric”.\nA coordinate symmetric body \\(B\\) is clearly determined by its restriction \\(B_+:=B\\cap\\Bbb R^n_+\\) to the positive orthant \\(\\Bbb R_+^n:=\\{x_i\\ge 0\\}\\). A set \\(B_+\\subset\\Bbb R^n_+\\) that can be written as such a restriction of a coordinate symmetric body is called a convex corner (also known as an anti-blocking body).\n\nConvex corners are in one-to-one correspondence with coordinate symmetric bodies. We prove the Mahler conjecture for coordinate symmetric bodies by formulating and proving a version on convex corners. For this we require a notion of polar duality that works for convex corners. If \\(B_+:= B\\cap\\Bbb R^n_+\\) then\n\\[B_+^* := \\{x\\in\\Bbb R^n_+\\mid \\langle x,y\\rangle\\le 1 \\text{ for all } y\\in B_+\\} = B^\\circ\\cap\\Bbb R^n_+.\\]\nSince a coordinate symmetric body consists of \\(2^n\\) identical copies of its convex corner and hence \\(\\vol(B)=2^n\\vol(B_+)\\), it is sufficient to prove the following:\n\nTheorem.  For each convex corner \\(B_+\\subset\\Bbb R^n_+\\) holds \\[M_+(B_+):=\\operatorname{vol}(B_+)\\operatorname{vol}(B_+^*) \\ge \\smash{\\frac1{n!}}.\\]\n\n\n\nProving the Mahler conjecture for convex corners\nThe following very short and elegant proof is due to Mathieu Meyer from his 1986 article Une caractérisation volumique de certains espaces normés. It was brought to my attention by Raman Sanyal, whose presentation of the proof I follow below.\nLet \\(B_i:=B_+\\cap\\{x_i=0\\}\\) and \\(B_i^*:=B_+^*\\cap \\{x_i=0\\}\\), and observe that those are themselves convex corners of dimension \\(n-1\\), and are duals of each other in the sense defined above. Define vectors \\(v,v^*\\in\\Bbb R^n\\) with components\n\\[v_i := \\frac{\\operatorname{vol}_{n-1}(B_i)}{n \\operatorname{vol}_{n}(B_+)},\\quad\\; v_i^* := \\frac{\\operatorname{vol}_{n-1}(B_i^*)}{n \\operatorname{vol}_{n}(B_+^*)}\\]\n(where from now on we put dimension subscripts on the volumes to avoid confusion). For a point \\(x\\in \\Bbb R_+^n\\) the inner product \\(\\langle x,v\\rangle\\) evaluates to\n\\[\n\\langle x,v\\rangle\n= \\sum_{i=1}^n x_i\\frac{\\operatorname{vol}_{n-1}(B_i)}{n \\operatorname{vol}_n(B_+)}\n= \\frac1{\\operatorname{vol}_{n}(B_+)} \\cdot \\sum_{i=1}^n \\underbrace{\\tfrac1n{x_i \\operatorname{vol}_{n-1}(B_i)}}_{\\operatorname{vol}_n(C_i)},\n\\]\nwhere \\(C_i\\) is the cone with base face \\(B_i\\) and apex at \\(x\\).\n\nIf \\(x\\in B_+\\) then the cones \\(C_i\\) have disjoint interiors and are contained in \\(B_+\\). In particular, the sum of their volumes is bounded by \\(\\operatorname{vol}_n(B_+)\\). Therefore \\(\\langle x,v\\rangle \\le 1\\) for all \\(x\\in B_+\\), and hence \\(v\\in B_+^*\\). By an analogous argument holds \\(v^*\\in B_+\\). Hence, we can conclude\n\\[\n\\begin{align}\n1\\,\\ge\\, &\\langle v,v^*\\rangle\n=  \\sum_{i=1}^n \\frac{\\operatorname{vol}_{n-1}(B_i)\\operatorname{vol}_{n-1}(B_i^*)}{n^2\\operatorname{vol}_n(B_+)\\operatorname{vol}_n(B_+^*)}\n= \\sum_{i=1}^n \\frac{M_+(B_i)}{n^2M_+(B_+)}.\n\\end{align}\n\\]\nBy rearranging and applying the induction hypothesis \\(M_+(B_i)\\ge 1/(n-1)!\\) we obtain\n\\[\nM_+(B_+) \\ge \\frac1{n^2} \\sum_{i=1}^n M_+(B_i) \\ge \\frac1{n^2} \\cdot\\frac{n}{(n-1)!} = \\frac1{n!}.\n\\]\nFrom this proof one can also extract the minimizers. Those turn out to be precisely the Hanner corners (that is, the positive corners of Hanner polytopes). Consequently, the minimizers among the coordinate symmetric bodies are indeed the Hanner polytopes.\n\n\nIt’s not about central symmetry anymore\nCoordinate symmetric bodies might seem a bit … restrictive. After all, they require quite some symmetry. However, as it turns out, once the problem is solved for convex corners, there is no longer any need to piece them together into a coordinate symmetric body to satisfy the Mahler conjecture. Even wilder, we don’t even need central symmetry anymore!\nA locally anti-blocking body is a convex body that if restricted to any orthant yields a convex corner (this is the official term, I don’t yet have a good idea for a more visually descriptive name).\n\nLocally anti-blocking bodies are a much richer family of bodie as compared to the highly symmetric coordinate symmetric bodies. In fact, they need not even be centrally symmetric! Still, they are closed under polar duality and share a number of nice properties with coordinate symmetric bodies.\nThe Mahler conjecture for locally anti-blocking bodies can be proven using the Cauchy-Schwarz inequality. For \\(\\sigma\\in\\{-1,+1\\}^n\\) write \\(\\Bbb R^n_\\sigma:=\\{x\\in\\Bbb R^n\\mid \\sigma_i x_i\\ge 0\\}\\). For a locally anti-blocking body \\(B\\) define\n\\[\n\\begin{align}\nw_\\sigma&:=\\operatorname{vol}( B\\cap \\Bbb R^n_\\sigma),\\quad\nw_\\sigma^*:=\\operatorname{vol}( B^\\circ\\cap \\Bbb R^n_\\sigma)\n\\end{align},\n\\]\nwhich are precisely the volumes of its \\(2^n\\) convex corners. Using the Cauchy-Schwarz inequality (CS) and the proof for convex corners \\((*)\\), we conclude\n\\[\n\\begin{align}\nM(B)\n= \\operatorname{vol}(B)\\operatorname{vol}(B^\\circ)\n&=\\Big(\\sum_\\sigma w_\\sigma\\Big)\\Big(\\sum_\\sigma w_\\sigma^*\\Big)\n\\\\&\\!\\overset{\\text{(CS)}}\\ge \\Big(\\sum_\\sigma \\sqrt{w_\\sigma w_\\sigma^*}\\Big)^{\\kern-1pt 2}\n\\overset{(*)}\\ge \\Big(\\frac{2^n}{\\sqrt{n!}}\\Big)^{\\kern-1pt 2} = \\frac{4^n}{n!}.\n\\end{align}\n\\]"
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    "title": "Do these 2-complexes embed in 4-space?",
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    "text": "Each \\(2\\)-dimensional CW complex can be embedded into 5-dimensional Euclidean space, but not necessarily into 4-dimensional space. In general, the problem of whether a given 2-complex embeds into \\(\\Bbb R^4\\) is believed to be very hard, potentially even undecidable. Still, I really really want to know whether my favorite 2-complexes embed!\n\nA 2-dimensional CW complex \\(\\mathcal X\\) is built from a topological graph (its 1-skeleton) together with 2-dimensional discs (its 2-cells) whose bondaries we glue to the graph along closed walks. Every 2-dimensional complex can be triangulated, so you lose nothing by thinking of simplicial complexes instead. Still, the constructions below go through easier if we allow for the flexibility of CW complexes. An embedding of \\(\\mathcal X\\) in 4-space is an injective map \\(\\mathcal X\\to\\Bbb R^4\\). You might assume a piecewise linear map to avoid thinking about topological pathologies.\nI will get to the heart of the matter right away, and deliver some background later on.\n\nBefore I tell you what I don’t know, let’s get accquainted with the task by looking at some standard examples. The complete 2-dimensional (simplicial) complex \\(\\mathcal K_n^2\\) is built from the complete graph \\(K_n\\) by attaching a 2-cell along each triangle (this is indeed a simplicial complex).\n\n\\(\\mathcal K_6^2\\) embeds in 4-space. Here is a quick construction using polytopes: \\(\\mathcal K_6^2\\) is the 2-skeleton of the 5-dimensional simplex \\(\\Delta_5\\). A Schlegel diagram of \\(\\Delta_5\\) is embedded in 4-space and contains an embedding of \\(\\mathcal K_6^2\\).\n\\(\\mathcal K_7^2\\) does not embed in 4-space. This is shown using the van Kampen obstuction (see below). Removing a single 2-simplex from \\(\\mathcal K_7^2\\) makes it embeddable. A nice argument for this was shown to me by Tâm Nguyên-Phan. I summarize it here.\n\n\nNow, let’s turn to some complexes for which the question of embeddability seems unanswered.\n\nThe join complex \\(\\mathcal J_{3,n}\\)\nStart by constructing the join of the complete graphs \\(K_3\\) and \\(K_n\\). This means add all possible edges between them. This yields \\(K_3\\sqcup K_{3,n}\\sqcup K_n\\simeq K_{n+3}\\). This will serve as the 1-skeleton of \\(\\mathcal J_{3,n}\\). Now we add the following 2-cells: for any four vertices \\(a,b\\in K_3\\) and \\(c,d\\in K_n\\) add the 2-cell along \\(abcd\\).\n\nI already know that \\(\\mathcal J_{3,4}\\) can be embedded in 4-space. An embedding is constructed from the embedding of \\(\\mathcal K_7^2\\) minus a 2-cell by a sequence of operations that preserve embeddability. If you are interested in details, let me refer you to my article with Agelos Georgakopoulos On 2-complexes embeddable in 4-space, and the excluded minors of their underlying graphs.\nThe first interesting case is \\(\\mathcal J_{3,5}\\), for which I don’t know the answer. Personally, I would like it to be not embeddable, but I really don’t know. The same holds for any \\(\\mathcal J_{3,n},n\\ge 5\\).\nHere are some modifications that do not diminish my interest in these complexes:\n\nfor \\(a,b\\in K_3\\) and \\(c,d\\in K_n\\) add further 2-cells along \\(abdc\\), \\(acbd\\) and \\(adbc\\).\nadd a 2-cell along the triangle in \\(K_3\\subset\\mathcal J_{3,n}\\).\nadd a cone over \\(K_n\\subset\\mathcal J_{3,n}\\). This means that we add a new vertex \\(x\\), add edges connecting it to all vertices of \\(K_n\\), and fill in all triangles that pass through \\(x\\) with 2-cells.\n\nI asked about this on MathOverflow without much success.\n\n\nThe long cycle complex \\(\\mathcal H_{2n-1}\\)\nThis is a short one.\nConstruct \\(\\mathcal H_{2n-1}\\) from the complete graph \\(K_{2n-1}\\) by attaching a 2-cell along each cycles of length \\(n\\) (alternatively, along each cycle of length at least \\(n\\)).\n\n\nSome background\nThe van Kampen obstruction is a standard technique to certify that a given 2-complex cannot be embedded into \\(\\Bbb R^4\\): you compute a number from \\(\\mathcal X\\) using a mechanical procedure, and if this number is non-zero, then you know that \\(\\mathcal X\\) cannot be embedded. However, if the computation yields zero (we say, the obstruction vanishes), you don’t learn anything about embeddability. We say, the van Kampen obstruction is incomplete. The mean fact on top: there exists an analogous obstruction for the embedding problem \\(n\\to2n\\) for each \\(n\\ge 1\\), and the obstruction fails to be complete only for \\(2\\to 4\\).\nThe standard example that shows that the van Kampen obstruction is incomplete is the Freedman-Krushkal-Teichner complex (or FKT complex for short). It has vanishing van Kampen obstruction, but does not embed in \\(\\Bbb R^4\\).\nI will not recall the van Kampen obstruction or the FKT complex. I do however want to mention that the FKT complex, while not super complicated in its construction, is not super simple either. I believe that the join complex or long cycle complex, if non-embeddable, have a chance to replace the FKT complex as the simplest known example for the van Kampen obstruction’s incompleteness.\nWhy does th van Kampen obstruction vanish for the above complexes? Even thought I wont recall the definition of the obstruction, I can give an intuition for why it vanishes. For computing the van Kampen onstruction one starts by enumerating all disjoint 2-cell pairs of \\(\\mathcal X\\), that is, all pairs \\((c_1,c_2)\\), where \\(c_1,c_2\\subseteq \\mathcal X\\) are 2-cells with \\(c_1\\cap c_2=\\varnothing\\). The computation then goes on working with this list, but we don’t need to know the details. Observe however that in all candidate complexes constructed above any two 2-cells intersect. That is, there are no disjoint 2-cell pairs. As a consequnce, the computation operates on an empty list, which results in the van Kampen obstruction being zero.\nLet me give yet another reasone for why the embeddability of the join complex in particular is of relevantce to me. Given a complex \\(\\mathcal X_\\Delta\\) containing a triangle \\(\\Delta\\) in its 1-skeleton with vertices, say, \\(x_1,x_2,x_3\\). Performing a \\(\\Delta\\mathrm Y\\)-transformations on \\(\\Delta\\) turns the complex into a new complex \\(\\mathcal X_{\\mathrm Y}\\) via the following steps:\n\ndelete the edges \\(x_1x_2\\), \\(x_2x_3\\) and \\(x_3x_1\\) of \\(\\Delta\\),\nadd a new vertex \\(y\\) as well as edges \\(yx_1\\), \\(yx_2\\) and \\(yx_3\\),\nreroute each 2-cell formerly attached along \\(x_i x_j\\) to now run along \\(x_i y x_j\\) instead.\n\nIn our article “On 2-complexes embeddable in 4-space, and the excluded minors of their underlying graphs”, Agelos Georgakopoulos and me posed the following conjecture:\n\nConjecture 3.14.  Suppose that the complex \\(\\mathcal X_\\Delta\\) is embeddable. Then the following are equivalent:\n\n\n\\(\\mathcal X_{\\mathrm Y}\\) is embeddable.\n\n\n\\(\\mathcal X_\\Delta\\) with a 2-cell attached along \\(\\Delta\\) is embeddable.\n\n\n\nWe proved the direction ii. \\(\\Longrightarrow\\) i., but the other direction remains open. If the join complex turns out embeddable, we can “improve it” it by adding a 2-cell along its triangle \\(K_3\\subset\\mathcal J_{3,n}\\). If this makes a difference for its embeddability, this would constitute a counterexample to the above conjecture."
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    "text": "“The stress-flex conjecture – a riddle in rigidity of coned polytopes” at the Kick-Off of the SPP 2458 [slides].\n\ncoming soon …\nIn the meantime, for slides and further information on my previous talks you can visit my old website at Warwick."
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    "text": "No time for reading long blog posts? Here are twitter length problems, questions, conjectures and thoughts.\n\n\n6 – Intrinsically triple linked graphs\nIs there a graph \\(G\\) so that for every embedding of \\(G\\) into 3-space there are three pairwise disjoint cycles of \\(G\\) that form a triple link, that is, any two of them are non-trivially linked?\n\n\n5 – Local minimizers of Mahler volume\nA centrally symmetric polytope is linearly discrete if it has only finitely many centrally symmetric realizations up to linear transformation. Is it true that the linearly discrete polytopes are precisely the local minimizers (among centrally symmetric polytopes) of the so-called Mahler volume \\(M(P):=\\operatorname{vol}(P)\\cdot\\operatorname{vol}(P^\\circ)\\)?\n\n\n4 – Embeddable complex?\nStart from \\(K_{2n-1}\\) and to each of its cycles of length \\(n\\) attach a 2-dimensional disk. The result is a 2-dimensional CW-complex. Does this complex embed in \\(\\Bbb R^4\\)?\nNow start from \\(K_{n+3}\\) with a distinguished decomposition into \\(K_3\\star K_n\\). Attach a disk along each cycle of length four that has one edge in \\(K_3\\) and one edge in \\(K_n\\). Does this CW complex embed in \\(\\Bbb R^4\\)?\n\n\n3 – Angles between random lines\nIf you choose two random lines (through the origin) uniformly, then the expected value of the angle beteen them is \\(1\\,\\mathrm{rad}\\). If you fix an orthonormal basis and choose two lines from them (repetitions allowed), then the expected angle is slightly larger, \\(\\pi/3\\approx 1.0472\\,\\mathrm{rad}\\). But what is the distribution of lines that gives the largest expected value for the angle? Is it the latter? (MO post)\n\n\n2 – Packing coloring the infinite path\nGiven a finite set \\(C\\) of “colors” together with a map \\(d:C\\to \\Bbb N\\) so that\n\\[\\delta(C):=\\sum_{c\\in C} \\frac1{d(c)+1} > \\sum_{n\\ge 0}\\frac1{2^n+1}\\approx 1.266.\\]\nDoes there exist a function \\(f:\\Bbb Z\\to C\\) so that for any two \\(i,j\\in\\Bbb Z\\) with \\(f(i)=f(j)=c\\) we have \\(|i-j|>d(c)\\)? This is known as a packing coloring of the infinite path, and it seems to exist once the density \\(\\delta(C)\\) is large enough.\n\n\n\n1 – Inscribed zonotopes\nInscribed zonotopes seem rare. Examples are the cube, prisms and the permutahedron. But how many different combinatorial types are there in 3D? The conjecture is that there are 17 distinct types + 1 infinite family of prisms. (relevant paper)"
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    "title": "Packing colorings and the mystery threshold 1.264499…",
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    "text": "\\[\\rlap{\\color{red}1}\\raise{-1pt}\\_\\, \\rlap{\\color{ForestGreen}2}\\raise{-1pt}\\_\\, \\rlap{\\color{blue}3}\\raise{-1pt}\\_\\, \\rlap{\\color{orange}4}\\raise{-1pt}\\_\\, \\rlap{\\color{violet}5}\\raise{-1pt}\\_\\, \\rlap{\\color{black}6}\\raise{-1pt}\\_\\, \\rlap{\\color{gray}7}\\raise{-1pt}\\_\\, \\rlap{\\color{purple}8}\\raise{-1pt}\\_\\, \\rlap{\\color{cyan}9}\\raise{-1pt}\\_\\,\\]\nIn a proper coloring of a graph any two vertices of the same color must be at distance at least two. In a packing coloring more general distance constraints can hold. For example, one might ask that any two vertices with the \\(i\\)-th color \\(c_i\\) are at distance \\(>i\\). Here is a packing coloring of the infinite path \\(P_\\infty\\) that is periodic in both directions:\n\\[\n\\dots\n\\rlap{\\color{red}1}\\raise{-1pt}\\_\\,\n\\rlap{\\color{blue}2}\\raise{-1pt}\\_\\,\n\\rlap{\\color{red}1}\\raise{-1pt}\\_\\,\n\\rlap{\\color{ForestGreen}3}\\raise{-1pt}\\_\\,\n\\rlap{\\color{red}1}\\raise{-1pt}\\_\\,\n\\rlap{\\color{blue}2}\\raise{-1pt}\\_\\,\n\\rlap{\\color{red}1}\\raise{-1pt}\\_\\,\n\\rlap{\\color{ForestGreen}3}\\raise{-1pt}\\_\\,\n\\rlap{\\color{red}1}\\raise{-1pt}\\_\\,\n\\dots\n\\]\nLets allow to have several colors with the same distance constraints. Here are two valid colorings using multiple colors.\n\\[\n\\dots\n\\rlap{\\color{red}1}\\raise{-1pt}\\_\\,\n\\rlap{\\color{blue}1}\\raise{-1pt}\\_\\,\n\\rlap{\\color{red}1}\\raise{-1pt}\\_\\,\n\\rlap{\\color{blue}1}\\raise{-1pt}\\_\\,\n\\rlap{\\color{red}1}\\raise{-1pt}\\_\\,\n\\rlap{\\color{blue}1}\\raise{-1pt}\\_\\,\n\\dots\n\\qquad\n\\dots\n\\rlap{\\color{red}2}\\raise{-1pt}\\_\\,\n\\rlap{\\color{blue}2}\\raise{-1pt}\\_\\,\n\\rlap{\\color{ForestGreen}2}\\raise{-1pt}\\_\\,\n\\rlap{\\color{red}2}\\raise{-1pt}\\_\\,\n\\rlap{\\color{blue}2}\\raise{-1pt}\\_\\,\n\\rlap{\\color{ForestGreen}2}\\raise{-1pt}\\_\\,\n\\dots\n\\]\nFor a general packing coloring we fix a pallette \\(\\boldsymbol c\\), that is, a finite or infinite sequence \\(c_1\\le c_2\\le c_3\\le\\dots\\) of natural numbers (which we call colors). We say that a graph \\(G\\) has a (packing) coloring using the pallette \\(\\boldsymbol c\\) if it can be colored using finitely many of the colors, so that any two vertices of color \\(c_i\\) have distance \\(>c_i\\).\nNote that a color \\(c_i\\) can appear at most at every \\((c_i+1)\\)-th vertex, hence, can have an asymptotic density of at most \\((c_i+1)^{-1}\\). We define the density of a pallette to be:\n\\[\\delta(\\boldsymbol c) := \\sum_i \\frac1{c_i+1}.\\]\nSince a proper packing coloring has asymptotic density exactly 1, we see that \\(\\delta(\\boldsymbol c)<1\\) implies that \\(\\boldsymbol c\\) does not packing color.\nIf \\(\\delta(\\boldsymbol c)<1\\), the pallette cannot color. In contrast, if \\(\\delta(\\boldsymbol c)>2\\), then the pallette does color. Somewhere in between there is a number \\(\\zeta\\) so that whenever \\(\\delta(\\boldsymbol c)>\\zeta\\), then \\(\\boldsymbol c\\) colors. We conjecture\n\\[\\zeta=\\sum_{i=1}^{\\infty} \\frac1{2^i+1} \\approx 1.264499...\\;.\\]\n\nConjecture  Each set of colors with density \\(\\ge \\zeta\\) does color the infinite path.\n\n\nTheorem  There is no finite packing coloring of the infinite path \\(P_\\infyt\\) using only the colors \\(2^n-1=1,3,7,15,...\\) where \\(n\\ge 1\\).\n\nProof.\nWe show that the path \\(P_n\\) of length \\(2^{n+1}-1\\) (that is, with this many vertices) cannot be colored using only color set \\(C_n:=\\{1,3,7,...,2^n-1\\}\\). This is clear for \\(n=1\\) (a path of length 3 cannot be colored using only color 1). Suppose then that \\(P_n,n\\ge 2\\) can be colored using \\(1,3,...,2^n-1\\). By induction hypothesis \\(C_{n-1}\\) is certainly not sufficient. So some vertex must have color \\(2^n-1\\). The vertex of color \\(2^n-1\\) splits \\(P_n\\) into two sub-paths, at least one of which is long enough to contain \\(P_{n-1}\\) but must be free of any occurance of color \\(2^n-1\\) and so must be colored using \\(C_{n-1}\\).This contradicts the induction hypothesis. \\(\\square\\)"
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    "title": "V-polytopes = H-polytopes",
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    "text": "There are two natural ways to define a convex polytope:\n\nas the convex hull of finitely many points (a V-polytope), and\nas the bounded intersection of finitely many half-spaces (an H-polytope).\n\nIt seems obvious that these describe the exact same class of objects. Though actually proving this then turns out less trivial than expected. For example, Günter Ziegler in his book “Lectures on Polytopes” goes the algorithmic rout via Fourier-Motzkin elimination. I always found this an overkill if one just wants to know that it is true. Below I present the shortest, most elementary and most direct proof that I was able to come up with.  It is short, but not super short. If you can think of a shorter or more direct proof, let me know. Also, let me know if there are gaps or parts that are actually more intricate than they seem.\n\nTheorem  V-polytopes \\(\\,=\\,\\) H-polytopes.\n\nProof\n\\(\\Longrightarrow\\):\nLet \\(P\\) be an H-polytope given as the bounded intersection of half-spaces \\(H_1,..., H_m\\). We want to find finitely many points \\(p_1,...,p_n\\) so that \\(P=\\operatorname{conv}(p_1,...,p_n)\\). We proceed by induction on the dimension \\(d\\) of the ambient space. The statement is clearly true for polytopes in \\(\\Bbb R\\) (aka. line segments). Let us therefore assume that \\(P\\subset\\Bbb R^d\\) for some \\(d\\ge 2\\).\nLet \\(\\partial H_i\\) denote the boundary hyperplane of \\(H_i\\).  Easy enough, the intersection \\(P_i:=P\\cap\\partial H_i\\) is an H-polytope in a lower-dimensional space. By induction hypothesis \\(P_i=\\operatorname{conv}(V_i)\\) for some finite point set \\(V_i\\subset\\Bbb R^d\\). We define \\(V=\\bigcup_i V_i\\), which is finite, and claim \\(P=\\operatorname{conv}(V)\\).\nOne direction is immediate: since \\(V\\subset P\\) we have \\(\\operatorname{conv}(V)\\subseteq \\operatorname{conv}(P)=P\\).\nFor the other direction fix a point \\(x\\in P\\). Consider some line \\(R:=\\{x+tv\\mid t\\in\\Bbb R\\}\\). Since \\(P\\) is convex and bounded, \\(R\\cap P\\) is a finite length line segment with end points \\(y_1,y_2\\). Easy enough, some of the halfspaces, say \\(H_1\\) and \\(H_2\\), satisfy \\(y_i\\in \\partial H_i\\), and hence, \\(y_i\\in P_i=\\operatorname{conv}(V_i)\\). We conclude \\(y_1,y_2\\in \\operatorname{conv}(V_1\\cup V_2)\\), and eventually\n\\[\nx\n\\in\\operatorname{conv}\\{y_1,y_2\\}\n\\subseteq\\operatorname{conv}(V_1\\cup V_2)\n\\subseteq\\operatorname{conv}(V).\n\\]\n\\(\\Longleftarrow\\):\n\nThis direction is especially easy if we assume some knowledge of polar duals: given a V-polytope \\(P=\\operatorname{conv}\\{p_1,..,p_n\\}\\), its polar dual is\n\\[P^\\circ:=\\big\\{x\\in\\Bbb R^d\\mid \\langle x,p_i\\rangle\\le 1\\text{ for all $i\\in\\{1,...,n\\}$}\\big\\}.\\]\nSuppose that \\(0\\in\\operatorname{int}(P)\\) (otherwise translate and restrict to the affine span). Some general properties of the polar map in this case are\n\n\\((P^\\circ)^\\circ=P\\)\n\\(P^\\circ\\) is bounded\n\\(P^\\circ\\) is contains the origin in its relative interior.\n\nWe already see that \\(P^\\circ\\) is an H-polytope defined by the half-spaces \\(H_i:=\\{\\langle x, p_i\\rangle \\le 1\\}\\). By the already established direction, \\(P^\\circ\\) is then also a V-polytope \\(P^\\circ=\\operatorname{conv}\\{q_1,...,q_m\\}\\). Thus, \\((P^\\circ)^\\circ=P\\) is an H-polytope defined by half-spaces \\(H_j^*:=\\{\\langle x, q_j\\rangle \\le 1\\}\\).\n\n\n\n\\(\\square\\)"
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    "text": "Today I hosted Anna Zamojska-Dzienio from the Warsaw University of Technology. She introduced me to barycentric algebras. Among other things, they allow us to see polytopes and face lattices as the same sort of object !  Intuitively, barycentric algebras are spaces in which one can take the convex combination of elements. I give here my own presentation of what she taught me. Since all of this is new to me there might be errors. Those are mine, not Anna’s.\n\nA barycentric algebra \\(\\mathcal B\\) is an algebra in the sense of “abstract algebra”, consisting of a ground set \\(B\\), and for each real number \\(r\\in(0,1)\\) a binary operation \\(\\underline{r}:B\\times B\\to B\\), subject to the following three axioms:\n\nidempotence : \\(\\;\\underline{r}(a,a)=a\\),\nskew-symmetry : \\(\\;\\underline{r}(a,b)=\\underline{r}^*(b,a)\\), where \\(r^*:=1-r\\),\nskew-associativity : using \\(r\\circ s:=(r^*s^*)^*=r+s-rs\\) it holds \\[\\underline r\\big(\\underline s(a,b),c\\big) = \\underline{\\smash{r\\circ s}}\\big(a,\\underline{\\tfrac{r}{r\\circ s}}(b,c)\\big).\\]\n\n\nThe two most instructive examples for barycentric algebras also represent the two extreme cases:\nConvex sets. A convex set \\(K\\subseteq\\Bbb R^n\\) can be identified as a barycentric algebra by setting\n\\[\\underline{r}(a,b):=(1-r)a+rb.\\]\nIn particular, polytopes are barycentric algebras. In some sense, polytopes are the finitely generated barycentric algebras (more details below).\nSemi-lattices. A (join) semi-lattice \\(\\mathcal L\\) with a join operation \\(\\vee\\) is a barycentric algebra under the operation\n\\[\\underline{r}(a,b):=a\\vee b.\\]\nIn particular, the face lattice \\(\\mathcal F(P)\\) of a polytope \\(P\\) is a barycentric algebra.\n\nA barycentric algebra homomorphism is a map \\(\\phi:\\mathcal A\\to\\mathcal B\\) with\n\\[\\phi\\big(\\underline{r}(a,b)\\big)= \\underline{r}\\big(\\phi(a),\\phi(b)\\big).\\]\n\nThe relation between polytopes and their face lattices is the following: the map \\(P\\to\\mathcal F(P)\\) that sends a point \\(x\\in P\\) to the unique face \\(\\sigma\\in\\mathcal F(P)\\) that contains \\(x\\) as an interior point, is a (barycentric algebra) homomorphism. And it is a distinguished homomorphism, as we will see shortly.\nThe relevant property that separates barycentric algebras of convex sets from the ones definde on semi-lattices turns out to be cancellativity:\n\\[\\underline{r}(a,b)=\\underline{r}(a,c) \\;\\implies\\; b=c.\\]\nConvex sets are cancellative; whereas semi-lattices are “the most non-cancellative” in the sense that they don’t have any non-trivial cancellative sub-algebras. Cancellativity is also a main reason for why we use \\(r\\in(0,1)\\) as opposed to \\(r\\in[0,1]\\): otherwise no barycentric algebra would be cancellative.\nBased on this, barycentric algebras are divided into three types:\n\ngeometric type are the cencellative barycentric algebras. They are intended to abstract convex sets.\ncombinatorial type have no non-trivial cancellative sub-algebras (in particular, are non-cancellative themselves). They are precisely the semi-lattices.\nmixed type fall into neither of the above classes.\n\nAn example of a mixed type is \\(\\Bbb R\\cup\\{\\infty\\}\\) with \\(\\underline{r}(x,\\infty)=\\infty\\) and normal convex combination on all other elements.\nThere exists a satisfying structure theory for barycentric algebras that separates the geometry from the combinatorics: each barycentric algebra can be subdivided into cancellative sub-algebras (the geometric parts), that when “contracted” leave a semi-lattice (the combinatorial part).\n\nTheorem   For a barycentric algebra \\(\\mathcal B\\) exists a unique homomorphism \\(\\phi:\\mathcal B\\to\\mathcal L\\) onto a semi-lattice \\(\\mathcal L\\) so that all preimages of elements \\(\\sigma\\in\\mathcal L\\) are cancellative. \n\nFor polytopes this distinguished semi-lattice is precisely the face-lattice (minus the empty face).\nThis provides us with an “abstract algebra” definition of face-lattice! A simplex is a finitely-generated free barycentric algebra; a polytope is a cancellative homomorphic image of a simplex (analogous to how polytopes are projections of simplices); and a face-lattice is a maximal “totally non-cancellative” homomorphic image of a polytope."
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    "text": "A zonotope is a polytope all whose faces are centrally symmetric. Equivalently (though not trivially so) a zonotope is the Minkowski sum of finitely many line segments. A polytope is inscribed if all its vertices lie on a sphere. Both zonotopes and inscribed polytopes are abundant. But it turns out that asking for both – that is, for inscribed zonotopes – is surprisingly restrictive and shows strong but not well-understood connections to reflection groups.\nIn 2D inscribed zonotopes are still rather boring: they are all of the following forms:\nIn total, only 17 inscribed zonotopes are known in dimension three (+ an infinite family of prisms), yet we can’t even show that there are finitely many (ignoring the prisms). But let’s move slowly.\nSome examples come to mind easily: the cube, prisms, and perhaps even the permutahedron.\n  \nAll those actually share a property much stronger than being inscribed: they are vertex-transitive, that is, their orthogonal symmetry group acts transitively on the vertices.\n…\nIn [1] I showed that the vertex-transitive zonotopes are precisely the \\(\\Gamma\\)-permutahedra, where \\(\\Gamma\\) is a finite reflection group.\nA vertex-transitive zonotope might have realizations that are inscribed but not vertex-transitive. For example, the \\(A_3\\)-zonotope can be realized as an inscribed zonotope with any of the following generators:\n\\[g_{ij}:=\\alpha_i\\alpha_j (\\alpha_i e_j-\\alpha_j e_i)\\]\n\nIn contrast, the \\(H_3\\)-zonotope can be realized as an inscribed zonotope in exactly one way, which is then vertex-transitive.\nSanyal and Manecke [1] noted that the projection of an inscribed zonotope along an edge direction yields again an inscribed zonotope. Starting from higher-dimensional \\(\\Gamma\\)-permutahedra this gives rise to many inscribed zonotopes of new combinatorial types – lets call them projected permutahedra. All in all this gives rise to 17 combinatorial types, for which they generated the graphics below.\n\nThey also gave the following diagram showing the realtions between projected permutahedra of different dimensions.\n\n\nConjecture  There are precisely 17 combinatorial types of inscribed zonotopes (+ one infinite family) in dimension three.\n\nThis seems very hard. In fact, it is now even known that the list of non-prismatic examples is finite, or that there is not another infinite family besides prisms. If the conjecture is true in dimension three one can feel somewhat confident that the above diagram also gives the precise number of (non-prismatic) inscribed zonotopes in higher dimensions, which would be: …\nTABLE (\\(d+2\\) in highe dimension)\nIt should be noted that all inscribed zonotopes are simple, that is, their edge graph is regular and the degree matches the dimension of the zonotope. As such, the hyperplane arrangement that corresponds to the zonotope (i.e. the arrangement whose normal vectors are the edge directions) is simplicial. Simplicial hyperplane arrangements form an exclusive club on their own. The “obvious” examples are the reflection arrangements. The most up-to-date list in dimension three is due to Grünbaum and Cuntz and list 93 arrangements + 3 infinite families. Sanyal and Manecke verified that only the quasi-reflection arrangements (the ones obtained by restriction of a reflection arrangements to a hyperplane) lead to inscribed zonotopes (on the way they verified that all the known arrangements are projectively unique!). They did however find that some arrangements lead to strange “zonotope-like” objects that are inscribed not in a sphere, but a different quadric (and thereby not convex)!\n…\nBesides that, inscribed zonotopes are closed under taking faces and projections along edge directions (or by iterating, along face directions).\n\n[1] S. Manecke, R. Sanyal, “Inscribable Fans II: Inscribed zonotopes, simplicial arrangements, and reflection groups”"
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    "text": "In my lecture today I proved that moving vertices radially in a coned framework preserves infinitesimal rigidity. This is a precursor to a number of important results, such as projective invariance of infinitesimal rigidity, or the equivalence of spherical and Euclidean infinitesimal rigidity. The most direct way to the result leads through a lengthy calculation where a lot of things can go wrong (and went wrong during the lecture). Here I redo the computation – carefully and in detail."
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    "title": "Coning framework",
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    "text": "Rigidity precursor\nSince this is my first post on framework rigidity, I should briefly recall the relevant notions of first-order theory. The reader comfortable with all these notions, can skip this section.\nA stress is a map \\(\\omega: E\\to\\Bbb R\\). We use the convention that \\(\\omega_{ij}=0\\) if \\(ij\\not\\in E\\) or if \\(i=j\\)."
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    "title": "Coned frameworks and radial moves",
    "section": "Coned frameworks",
    "text": "Coned frameworks\nA framework \\((G,\\boldsymbol p)\\) is coned if the graph \\(G\\) has a dominating vertex, that is, a vertex, say \\(0\\in V\\), that is adjacent to all other vertices.\nA radial move in a coned framework slides a vertex (that is not the cone vertex and does not lie on the cone vertex) along the ray that connects it to the cone vertex. Assume for simplicity that \\(p_0=0\\); then a radial move is more precisely given by \\(p_i\\mapsto r_i p_i\\) for some \\(r_i\\not=0\\).\n\nTheorem   Radial moves preserve the rank of the rigidity matrix, and in particular, preserve first-order rigidity and flexibility. \n\nProof.\nLet \\((G,\\boldsymbol p)\\) be a coned framework, and \\((G,\\hat{\\boldsymbol p})\\) obtained from it by radial moves. As bfore, we assum \\(p_0=0\\) and \\(\\hat p_i=r_i p_i\\) for \\(r_i\\not=0\\).\nPerhaps ones first attempt would be to construct a one-to-one correspondence between infinitesimal flexes of \\((G,\\boldsymbol p)\\) and \\((G,\\hat{\\boldsymbol p})\\). While possible, there is no nice explicit description of this. The trick is to instead construct a one-to-one correspondence between (equilibrium) stresses. …\nWhile there are indeed canonical maps between the spaces of infinitesimal flexes \\(\\ker R(G,\\boldsymbol p)\\) and \\(\\ker R(G,\\hat{\\boldsymbol p})\\), it turns out that they are rather complicated to describe explicitly. Much easier it is to construct maps between the spaces of equilibrium stresses \\(\\mathop{\\mathrm{coker}}R(G,\\boldsymbol p)\\) and \\(\\mathop{\\mathrm{coker}}R(G,\\hat{\\boldsymbol p})\\). Clearly, either map would witnesses the equality in rank of the rigidity matrices.\nSuppose then that we are given an (equilibrium) stress \\(\\boldsymbol\\omega:E\\to\\Bbb R\\) of \\((G,\\boldsymbol p)\\), and let us try to construct (one-to-one) an equilibrium stress \\(\\hat{\\boldsymbol\\omega}:E\\to\\Bbb R\\) for the radially rescaled framework \\((G,\\hat{\\boldsymbol p})\\).\nWe make the natural guess \\(\\hat\\omega_{ij}:=\\omega_{ij}/(r_ir_j)\\) for the non-cone edges \\(ij\\in E\\). The corresponding natural guess \\(\\hat\\omega_{ij}:=\\omega_{i0}/r_i^2\\) for cone edges is actually not correct, and the correct choice is not really guessable. Instead we derive it by enforcing the equilibrium condition \\((*)\\) at vertex \\(i\\not=0\\):\n\\[\\begin{align}\n    0\\;\n    &\\overset!=\n    \\sum_{j} \\hat\\omega_{ij} (\\hat p_j-\\hat p_i)\n    \\\\&=\n    \\sum_{\\mathclap{j\\not=0}} \\hat\\omega_{ij} (\\hat p_j-\\hat p_i) + \\smash{\\overbrace{\\hat\\omega_{i0} (\\hat p_0-\\hat p_i)}^{-\\hat\\omega_{i0}\\hat p_i\\mathrlap{\\text{\\color{lightgray}$\\;\\;\\leftarrow$ we used $p_0=0$}}}}\n    \\\\&=\n    \\sum_{\\mathclap{j\\not=0}} \\frac{\\omega_{ij}}{r_i r_j} (r_jp_j- r_ip_i) -\\hat\\omega_{i0}r_ip_i\n    \\\\&=\n    \\tfrac1{r_i} \\sum_{\\mathclap{j\\not=0}} \\omega_{ij} p_j + \\Big(\\sum_{\\mathclap{j\\not=0}} \\omega_{ij} \\tfrac1{r_j} \\Big)p_i -\\hat\\omega_{i0}r_ip_i\n    \\\\&=\n    \\tfrac1{r_i} \\underbrace{\\sum_{\\mathclap{j\\not=0}} \\omega_{ij} (p_j - p_i)}_{-\\omega_{i0}(p_0-p_i)\\,=\\, \\omega_{i0}p_i \\mathrlap{\\text{\\color{lightgray}$\\;\\;\\leftarrow$ we used $(*)$ for $\\boldsymbol\\omega$ at vertex $i$}}} +\n    \\Bigg(\\sum_{\\mathclap{j\\not=0}} \\omega_{ij} \\Big(\\tfrac1{r_j}-\\tfrac1{r_i}\\Big) - \\hat\\omega_{i0}r_i \\Bigg)p_i\n    \\\\&=\n    \\Bigg(\\tfrac1{r_i}\\omega_{i0} + \\sum_{\\mathclap{j\\not=0}} \\omega_{ij} \\Big(\\tfrac1{r_j}-\\tfrac1{r_i}\\Big) - \\hat\\omega_{i0}r_i\\Bigg)p_i\n\\end{align}\\]\nThus, if \\(p_i\\not= 0\\), we necessarily have to set\n\\[\n\\hat\\omega_{i0}:= \\frac{\\omega_{i0}}{r_i^2} + \\sum_{j\\not=0} \\omega_{ij} \\Big(\\tfrac1{r_i}-\\tfrac1{r_j}\\Big).\n\\]\n(and if \\(p_i=0\\), then the we can set \\(\\hat\\omega_{i0}\\) arbitrarily). This is not far from our ansatz: it is the natural guess \\(\\omega_{i0}/r_i^2\\) plus some reasonably nice correction term.\nIt remains to check that \\(\\hat{\\boldsymbol\\omega}\\) satisfies the equilibrium condition \\((*)\\) also at the cone vertex. But this can be inferred without much computation. For the students in my course I argue using some terminology (for everyone else, see the self-contained argument below): the stress \\(\\hat{\\boldsymbol\\omega}\\) gives rise to a resolvable load \\(f_i:=-\\sum_j \\hat\\omega_{ij}(\\hat p_j-\\hat p_i)\\). By construction, \\(f_i=0\\) for all \\(i\\not=0\\). But resolvable loads are equilibrium loads, in particular, \\(\\sum f_i=0\\), and so \\(f_0=0\\) follows as well.\nThis can be argument without the slang: if a stress \\(\\boldsymbol\\omega\\) satisfies the equilibrium condition \\((*)\\) at all vertices \\(i\\not=0\\), then it can be inferred to hold at \\(i=0\\) as well like this (here \\(\\boldsymbol p\\) and \\(\\boldsymbol\\omega\\) are some points and stress unrelated to the setting above):\n\\[\\begin{align}\n\\sum_i \\omega_{i0} (p_0-p_i)\n&= \\sum_{i,j} \\omega_{ij} (p_j - p_i) - \\sum_{j\\not=0}\\overbrace{\\sum_{i} \\omega_{ij} (p_j - p_i)}^{=0}\n\\\\&= \\sum_{i<j} \\omega_{ij} (p_j-p_i) + \\sum_{i>j}\\omega_{ij}(p_j-p_i)\n\\\\&= \\sum_{i<j} \\omega_{ij} (p_j-p_i) + \\sum_{j>i}\\omega_{ji}(p_i-p_j) \\mathrlap{\\quad\\text{\\small\\color{lightgray}(we renamed $i$ and $j$)}}\n\\\\&= \\sum_{i<j} \\omega_{ij} (p_j-p_i) - \\sum_{i<j}\\omega_{ij}(p_j-p_i)\n=0\n\\end{align}\\]\nWe now know that \\(\\hat{\\boldsymbol\\omega}\\) is an equilibrium stress of \\((G,\\hat{\\boldsymbol p})\\). Also, it is a linear transformation of \\(\\boldsymbol\\omega\\), and clearly an invertible one (use \\(1/r_i\\) instead of \\(r_i\\) for the inverse). Hence we described a map between the equilibrium stress spaces of \\((G,\\boldsymbol p)\\) and \\((G,\\hat{\\boldsymbol p})\\), which proves the result.\n\n\\(\\square\\)"
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    "text": "Rigidity primer\nSince this is my first post on framework rigidity, I should briefly recall the relevant notions of first-order theory. The experienced reader can skip this section.\nA first-order motion (also infinitesimal motion) is a map \\(\\dot{\\boldsymbol p}:V\\to\\Bbb R^d\\) that satisfies\n\\[(*)\\quad \\langle p_j-p_i, \\dot p_j-\\dot p_i\\rangle = 0, \\quad\\text{for all $ij\\in E$}.\\]\nSome first-order motions always exist: infinitesimal translations and rotations. Those are said to be trivial and can be neatly defined as being of the form \\(\\dot {\\boldsymbol p} = S\\boldsymbol p + v\\) where \\(S\\in\\Bbb R^{d\\times d}\\) is skew-symmetric and \\(v\\in\\Bbb R^d\\). A first-order motion that is not trivial is called a first-order flex. If a framework as a first-order flex, it is called first-order flexible, otherwise first-order rigid.\nThe equations \\((*)\\) form a linear system and there is a matrix \\(R(G,\\boldsymbol p)\\in\\Bbb R^{E\\times dV}\\) that encapsulates these conditions so that \\(\\dot{\\boldsymbol p}\\) is a first-order motion if and only if \\(R(G,\\boldsymbol p)\\dot{\\boldsymbol p}=0\\). This matrix is called the rigidity matrix of the framework, and is at the core of the first-order theory of rigidity.\nA stress is a map \\(\\boldsymbol\\omega:E\\to\\Bbb R\\). It is often convenient to set \\(\\omega_{ij}=0\\) whenever \\(ij\\not\\in E\\) or if \\(i=j\\). An equilibrium stress is a stress that satisfies\n\\[(**)\\quad \\sum_{\\mathclap{j}}\\omega_{ij}(p_j-p_i)=0,\\quad\\text{for all $i\\in V$}.\\]\nThe rigidity matrix \\(R(G,\\boldsymbol p)\\) of a framework captures both of the above linear systems.\n\n\\(\\dot{\\boldsymbol p}\\) is first-order flex if and only if \\(\\dot{\\boldsymbol p}\\in\\ker R(G,\\boldsymbol p)\\).\n\\(\\boldsymbol\\omega\\) is an equilibrium stress if and only if \\(\\boldsymbol\\omega\\in\\mathop{\\mathrm{coker}}R(G,\\boldsymbol p)=\\ker R^\\top(G,\\boldsymbol p)\\)."
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    "title": "Coned frameworks and radial moves",
    "section": "Rigidity crash course",
    "text": "Rigidity crash course\nSince this is my first post on framework rigidity, I will briefly introduce the relevant notions of first-order rigidity. The knowledgeable reader can skip this section.\nA first-order motion (also infinitesimal motion) is a map \\(\\dot{\\boldsymbol p}:V\\to\\Bbb R^d\\) that satisfies\n\\[(*)\\quad \\langle p_j-p_i, \\dot p_j-\\dot p_i\\rangle = 0, \\quad\\text{for all $ij\\in E$}.\\]\nSome first-order motions are said to be trivial because they exist irrespective of the exact framework: infinitesimal translations and rotations. They can be written in the form \\(\\dot {\\boldsymbol p} = S\\boldsymbol p + v\\) where \\(S\\in\\Bbb R^{d\\times d}\\) is skew-symmetric and \\(v\\in\\Bbb R^d\\). A first-order motion that is not trivial is called a first-order flex. A framework that has a first-order is said to be first-order flexible, and first-order rigid otherwise.\n\nThe equations \\((*)\\) form a linear system and so there is a matrix \\(R(G,\\boldsymbol p)\\in\\Bbb R^{E\\times dV}\\) whose rows correspond to the equations \\((*)\\). That is, \\(\\dot{\\boldsymbol p}\\) is a first-order motion if and only if \\(R(G,\\boldsymbol p)\\dot{\\boldsymbol p}=0\\). This matrix is called the rigidity matrix of the framework, and it is, in my opinion, the cleanest way to unify the various approaches to first-order theory.\nA stress is a map \\(\\boldsymbol\\omega:E\\to\\Bbb R\\). It is often convenient to set \\(\\omega_{ij}=0\\) whenever \\(ij\\not\\in E\\) or if \\(i=j\\). An equilibrium stress is a stress that satisfies\n\\[(**)\\quad \\sum_{\\mathclap{j}}\\omega_{ij}(p_j-p_i)=0,\\quad\\text{for all $i\\in V$}.\\]\nThe rigidity matrix \\(R(G,\\boldsymbol p)\\) of a framework captures both of the above linear systems.\n\n\\(\\dot{\\boldsymbol p}\\) is first-order flex if and only if \\(\\dot{\\boldsymbol p}\\in\\ker R(G,\\boldsymbol p)\\).\n\\(\\boldsymbol\\omega\\) is an equilibrium stress if and only if \\(\\boldsymbol\\omega\\in\\mathop{\\mathrm{coker}}R(G,\\boldsymbol p)=\\ker R^\\top(G,\\boldsymbol p)\\)."
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    "title": "Coned frameworks and radial moves",
    "section": "Coned frameworks and radial moves",
    "text": "Coned frameworks and radial moves\nA framework \\((G,\\boldsymbol p)\\) is coned if the graph \\(G\\) has a dominating vertex, that is, a vertex that is adjacent to all other vertices. We shall denote this cone vertex by \\(*\\in V\\).\nA radial move in a coned framework slides vertices (other than the cone vertex) along the rays that connect them to the cone vertex. If we assume for simplicity that \\(p_*=0\\), then a radial move is given by \\(p_i\\mapsto r_i p_i\\) for \\(r_i\\not=0\\).\n\n\nTheorem   Radial moves preserve first-order rigidity and flexibility. More strongly, radial moves preserve the rank of the rigidity matrix. \n\nProof.\nLet \\((G,\\boldsymbol p)\\) be a coned framework, and \\((G,\\hat{\\boldsymbol p})\\) obtained from it by radial moves. As before, we assum \\(p_*=0\\) and \\(\\hat p_i=r_i p_i\\) for \\(r_i\\not=0\\).\nWith the primary goal to show that radial moves preerves first-order rigidity, which is defined in terms of first-order flexes, ones first attempt is perhaps to construct an explicit one-to-one correspondence between first-order flexes of \\((G,\\boldsymbol p)\\) and \\((G,\\hat{\\boldsymbol p})\\). This turns out quite tedious. Instead, following a previous comment, the trick is to construct a one-to-one correspondence between equilibrium stresses. Since\n\\[\\operatorname{rank}R(G,\\boldsymbol p)=\\operatorname{rank}R(G,\\boldsymbol p)^\\top=E-\\!\\overbrace{\\,\\ker R(G,\\boldsymbol p)^\\top\\,}^{\\mathclap{\\text{space of equilibrium stresses}}}\\!,\\]\nthis also shows that the rank of the rigidity matrix is preserved.\n\nSuppose then that we are given an (equilibrium) stress \\(\\boldsymbol\\omega:E\\to\\Bbb R\\) of \\((G,\\boldsymbol p)\\), and let us try to construct an equilibrium stress \\(\\hat{\\boldsymbol\\omega}:E\\to\\Bbb R\\) for the radially rescaled framework \\((G,\\hat{\\boldsymbol p})\\), so as to obtain a one-to-one map between the stress spaces. We shall adopt the convention that \\(\\omega_{ij}=0\\) whenever \\(ij\\not\\in E\\) or \\(i=j\\), which makes many expressions less cluttered.\nWe make the natural guess \\(\\hat\\omega_{ij}:=\\omega_{ij}/r_ir_j\\) for the non-cone edges \\(ij\\in E\\). The corresponding natural guess \\(\\hat\\omega_{i*}:=\\omega_{i*}/r_i^2\\) for cone edges does actually not work, and the correct choice is hardly guessable. Instead we derive it by enforcing the equilibrium condition \\((**)\\) at vertex \\(i\\not=*\\):\n\\[\\begin{align}\n    0\\;\n    &\\overset!=\n    \\sum_{j} \\hat\\omega_{ij} (\\hat p_j-\\hat p_i)\n    \\\\&=\n    \\sum_{\\mathclap{j\\not=*}} \\hat\\omega_{ij} (\\hat p_j-\\hat p_i) + \\smash{\\overbrace{\\hat\\omega_{i*} (\\hat p_*-\\hat p_i)}^{-\\hat\\omega_{i*}\\hat p_i\\mathrlap{\\text{\\color{lightgray}$\\;\\;\\leftarrow$ we used $p_*=0$}}}}\n    \\\\&=\n    \\sum_{\\mathclap{j\\not=*}} \\frac{\\omega_{ij}}{r_i r_j} (r_jp_j- r_ip_i) -\\hat\\omega_{i*}r_ip_i\n    \\\\&=\n    \\tfrac1{r_i} \\sum_{\\mathclap{j\\not=*}} \\omega_{ij} p_j + \\Big(\\sum_{\\mathclap{j\\not=*}} \\omega_{ij} \\tfrac1{r_j} \\Big)p_i -\\hat\\omega_{i*}r_ip_i\n    \\\\&=\n    \\tfrac1{r_i} \\underbrace{\\sum_{\\mathclap{j\\not=*}} \\omega_{ij} (p_j - p_i)}_{-\\omega_{i*}(p_*-p_i)\\,=\\, \\omega_{i*}p_i \\mathrlap{\\text{\\color{lightgray}$\\;\\;\\leftarrow$ we used $(**)$ for $\\boldsymbol\\omega$ at vertex $i$}}} +\n    \\Bigg(\\sum_{\\mathclap{j\\not=*}} \\omega_{ij} \\Big(\\tfrac1{r_j}-\\tfrac1{r_i}\\Big) - \\hat\\omega_{i*}r_i \\Bigg)p_i\n    \\\\&=\n    \\Bigg(\\tfrac1{r_i}\\omega_{i*} + \\sum_{\\mathclap{j\\not=*}} \\omega_{ij} \\Big(\\tfrac1{r_j}-\\tfrac1{r_i}\\Big) - \\hat\\omega_{i*}r_i\\Bigg)p_i\n\\end{align}\\]\nThus, if \\(p_i\\not= 0\\), we necessarily have to set\n\\[\n\\hat\\omega_{i*}:= \\frac{\\omega_{i*}}{r_i^2} + \\sum_{j\\not=*} \\omega_{ij} \\Big(\\tfrac1{r_i}-\\tfrac1{r_j}\\Big).\n\\]\n(and if \\(p_i=0\\), then the we can set \\(\\hat\\omega_{i*}\\) arbitrarily). This is not far from our ansatz: it is the natural guess \\(\\omega_{i*}/r_i^2\\) plus some reasonably nice correction term.\nIt remains to check that \\(\\hat{\\boldsymbol\\omega}\\) satisfies the equilibrium condition \\((**)\\) also at the cone vertex. But this can be inferred without much computation. For the students in my course I argue using some terminology (for everyone else, see the self-contained argument below): the stress \\(\\hat{\\boldsymbol\\omega}\\) gives rise to a resolvable load \\(f_i:=-\\sum_j \\hat\\omega_{ij}(\\hat p_j-\\hat p_i)\\). As shown above, \\(f_i=0\\) for all \\(i\\not=*\\). But resolvable loads are equilibrium loads, in particular, \\(\\sum f_i=0\\). Thus, \\(f_*=0\\) follows as well.\nThis can be argued without the slang: if a stress \\(\\hat{\\boldsymbol\\omega}\\) satisfies the equilibrium condition \\((**)\\) at all vertices \\(i\\not=*\\), then it can be inferred to hold at \\(i=*\\) as well: \n\\[\\begin{align}\n\\sum_i \\hat\\omega_{i*} (\\hat p_*-\\hat p_i)\n&= \\sum_{i,j} \\hat \\omega_{ij} (\\hat p_j - \\hat p_i) - \\sum_{j\\not=*}\\overbrace{\\sum_{i} \\hat \\omega_{ij} (\\hat p_j - \\hat p_i)}^{=0}\n\\\\&= \\sum_{i<j} \\hat \\omega_{ij} (\\hat p_j-\\hat p_i) + \\sum_{i>j}\\hat \\omega_{ij}(\\hat p_j-\\hat p_i)\n\\\\&= \\sum_{i<j} \\hat \\omega_{ij} (\\hat p_j-\\hat p_i) + \\sum_{j>i}\\hat \\omega_{ji}\\smash{\\overset{\\mathrlap{\\;\\text{\\color{lightgray}(we renamed $i$ and $j$)}}}(}\\hat p_i-\\hat p_j)\n\\\\&= \\sum_{i<j} \\hat \\omega_{ij} (\\hat p_j-\\hat p_i) - \\sum_{i < j}\\hat \\omega_{ij}(\\hat p_j-\\hat p_i)\n=0\n\\end{align}\\]\nWe have conclusively shown that \\(\\hat{\\boldsymbol\\omega}\\) is an equilibrium stress for \\((G,\\hat{\\boldsymbol p})\\). Also, \\(\\hat{\\boldsymbol\\omega}\\) is a linear expression in terms of \\(\\boldsymbol\\omega\\), and clearly an invertible one (use \\(1/r_i\\) instead of \\(r_i\\) for the inverse). Hence we described a linear isomorphism between the equilibrium stress spaces of \\((G,\\boldsymbol p)\\) and \\((G,\\hat{\\boldsymbol p})\\). This concludes the proof.\n\n\\(\\square\\)"
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    "text": "First-order rigidity crash course\nSince this is my first post on framework rigidity, I will briefly introduce the relevant notions of first-order rigidity. The informed reader may skip this section.\nA first-order motion (also infinitesimal motion) is a map \\(\\dot{\\boldsymbol p}:V\\to\\Bbb R^d\\) that satisfies\n\\[(*)\\quad \\langle p_j-p_i, \\dot p_j-\\dot p_i\\rangle = 0, \\quad\\text{for all $ij\\in E$}.\\]\nThe intuition is that moving vertices in the directions of a first-order motion does not violate any edge length constraint in the first order. Each (differentiable) motion of a framework gives rise to a first-order motion by differtiating at time \\(t=0\\); but not all first-order motions need to correspond to “actual” motions in this way (and this is precisely what makes rigidity different from first-order rigidity). Some first-order motions are called trivial because they exist irrespective of the details of the framework: infinitesimal translations and rotations. They can be written in the form \\(\\dot {\\boldsymbol p} = S\\boldsymbol p + v\\) where \\(S\\in\\Bbb R^{d\\times d}\\) is skew-symmetric and \\(v\\in\\Bbb R^d\\). A first-order motion that is not trivial is called a first-order flex. A framework that has a first-order flex is said to be first-order flexible. It is called first-order rigid if it has none. First-order rigidity implies actual rigidity, whereas first-order flexibility does not always imply actual flexibility.\nFor the linear system \\((*)\\) we can write down a matrix \\(R(G,\\boldsymbol p)\\in\\Bbb R^{E\\times dV}\\) whose rows correspond to the individual equations. Then \\(\\dot{\\boldsymbol p}\\) is a first-order motion if and only if \\(R(G,\\boldsymbol p)\\dot{\\boldsymbol p}=0\\), or equivalenty, if \\(\\dot{\\boldsymbol p}\\in\\ker R(G,\\boldsymbol p)\\). This matrix is called the rigidity matrix of the framework. It has the following structure:\n\nThe rigidity matrix is a useful way of packaging the linear details of a framework. Essentially all of first-order theory can be expressed rather elegantly using it. For example, a framework (with \\(V\\ge d+1\\) vertices) is first-order rigid if and only if \\(\\operatorname{corank}R(G,\\boldsymbol p)={d+1\\choose 2}\\). Moreover, the rank of the rigidity matrix can be seen as a more fine-grained measure of rigidity, or of “how far away from rigid” a framework is (in the first-order sense).\nThe other first-order notion that is often useful is a stress, which is a map \\(\\boldsymbol\\omega:E\\to\\Bbb R\\).  An equilibrium stress is a stress that satisfies\n\\[(**)\\quad \\sum_{\\mathclap{j:ij\\in E}}\\omega_{ij}(p_j-p_i)=0,\\quad\\text{for all $i\\in V$}.\\]\nThe physical interpretation of stress is as an internal force in an edge that tries to expand or contract the edge by pushing and pulling on its end vertices along the edge direction. An equilibrium stress is then a stress where at each vertex these forces cancel out. It turns out that \\(\\boldsymbol\\omega\\) is an equilibrium stress if and only if \\(R^\\top\\!(G,\\boldsymbol p)\\boldsymbol\\omega=0\\), or equivalently, if \\(\\boldsymbol\\omega\\in\\ker R^\\top\\!(G,\\boldsymbol p)=\\mathop{\\mathrm{coker}}R(G,\\boldsymbol p)\\).\n\nSince first-order motions and stresses appear as the kernel and cokernel of the rigidity matrix respectively, one can use either to study its rank, and hence, the first-order rigidity of \\((G,\\boldsymbol p)\\). In fact, even though one usually aims to make a statement about first-order flexes, it often turs out more convenient to argue about stresses. One reason is that stresses have no equivalent of “trivial motions” one needs to account for. We will see an example of this when we consider coned frameworks."
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    "text": "Finger moves are the elementary operations used to transform any embedding of a graph or CW complex into any other embedding. Keeping track of how a finger move change intersections between edges and cells gives rise to well-known obstructions to planarity, linklessness, and also embeddability into higher dimensional spaces. I recall here this elegant idea, and also show when the obstructions is preserved under \\(\\Delta\\mathrm Y\\)-transformations.\n\n\nPlanar graphs\nFor a warmup, let’s see how finger moves let us conclude quickly that \\(K_5\\) is not a planar graph. Observe that in the following drawing there are five crossings between (non-adjacent) edges.\n\nAny other drawing of \\(K_5\\) can be obtained from this drawing by moving around vertices and edges, potentially pulling edges over vertices and edges over other edges. The technical term is a finger move, that is, pulling an edge over a vertex, thereby changing the edges with which it intersects (pulling an edge over another edge can be seen as pulling it over a vertex of degree two). The key observation is that since each vertex in \\(K_5\\) is of degree four, a finger move preserves the parity of the total number of intersections in the drawing.\n\nNote that we only count intersections between non-adjacent edges (in fact, otherwise it does not work).\nSince this total number of intersections was odd to begin with (it was five), it can never become zero. Hence, \\(K_5\\) has no drawing without crossing edges.\nAlmost the same argument works for \\(K_{3,3}\\). Here is a drawing of \\(K_{3,3}\\) with seven crossings.\nThe problem is that in \\(K_{3,3}\\) all vertices have degree three, and so a finger move changes parity. We can apply a trick: add new edges so as to make all degrees even, but preserve the potential planarity of the graph. Preserving planarity is easy if we just add edges parallel to edges that are already there. Alterantively one can see this has adding a weight to edges so that the weighted degree is always even. Note that we then also have to count the crossings in the initial drawing with multiplicity: it is nine.\nThe much more interesting observation, tha we wont go into here, is that even parity for the total intersection already guarantees that the graph is planar!\nThe ida of finger moves generalizes splendidly! The one thing one needs to come up with is the think one want to count and show to be odd.\n\n\nLinks and knots\nAn analogous trick shows that \\(K_6\\) is intrisnically linked and \\(K_7\\) is intrisically knotted, onyl that here finger moves pull edges over edges, and we need to count something more compliated than edge intersections.\n\nObserve that the following embedding of \\(K_6\\) has a single pair of linked cycles of linking number one.\nObserve that crossing two edges over each other changes the sum of linking numbers of all disjoint cycle pairs in \\(K_6\\) by an even number.\nConclude that each spacial embedding of \\(K_6\\) must have two cycles of odd linking number, hence, must be linked.\n\nLikewise\n\nObserve that the following embedding of \\(K_7\\) has a single knotted cycle whose \\(\\operatorname{arf}\\) invariant is one.\nObserve that crossing two edges over each other changes the sum of \\(\\operatorname{arf}\\) of all cycles by an even number.\nConclude that each spacial embedding of \\(K_7\\) must have a cycle of odd \\(\\operatorname{arf}\\) invariant, hence, must be knotted.\n\n\n\nThe van Kampen obstruction\nThe idea of the finger moves generalizes to embedding into higher dimensions. We consider there embedding 2-dimensional complexes into \\(\\Bbb R^4\\).\n\nstart with some arbitrary embedding of \\(X\\) into \\(\\Bbb R^4\\)\ncount the total number of intersections between disjoint pairs of 2-cells and observe that it is odd.\nshow that when moving a 2-cell across an edge that the total number of intersection changes by an even number, that is, parity is preserved.\nConclude that no embedding can have intersection number zero.\n\nThis is known as the van Kampen onstruction, first introduced by … van Kampen in 19…\nThis gives an easy way to see that, for example, the complete 2-dimensional complex \\(\\mathcal K_7^{(2)}\\) does not embedd in \\(\\Bbb R^4\\) (see [here] for a neat construction of an embedding of \\(\\mathcal K_7^{(2)}\\) with a single intersection between 2-cells).\nThere are more modern ways to define the van Kampen obstruction via homology groups of deleted products. There are also straightforward ways to generlize these obstructions to even higher dimensions. The mean fact is that this obstruction turns out to be complete, that is, even van Kampen obstruction implies embeddability, if and only if \\(d\\not= 4\\).\n\n\n\\(\\Delta\\mathrm Y\\)-transformations\n…\n\nTheorem.   \\(\\Delta\\mathrm Y\\)-transformations preserve the parity of total intersections of 2-cells. \n\nThe analogous statement is true for planar graphs, which we leave as an exercise to the reader.\nProof.\n…\n\\(\\square\\)"
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    "title": "Coned frameworks and radial moves",
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    "text": "In my lecture today I proved that moving vertices radially in a coned framework preserves first-order rigidity. This is a precursor to a number of important results, such as projective invariance of first-order rigidity, or the equivalence of generic rigidity on the sphere and Euclidean space. The most direct way to the result leads through a lengthy calculation where a lot of things can go wrong (and went wrong during the lecture). Here I redo the computation – carefully and in detail."
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    "section": "First-order rigidity crash course",
    "text": "First-order rigidity crash course\nSince this is my first post on framework rigidity, I will briefly introduce the relevant ideas of the first-order theory of rigidity. The informed reader may skip this section.\nA first-order motion (also infinitesimal motion) is a map \\(\\dot{\\boldsymbol p}:V\\to\\Bbb R^d\\) that satisfies\n\\[(*)\\quad \\langle p_j-p_i, \\dot p_j-\\dot p_i\\rangle = 0, \\quad\\text{for all $ij\\in E$}.\\]\nThe intuition is that moving vertices in the directions of a first-order motion does not violate any edge length constraint in the first order. Each (differentiable) motion of a framework gives rise to a first-order motion by differtiating at time \\(t=0\\); but not all first-order motions need to correspond to “actual” motions in this way (and this is precisely what makes rigidity different from first-order rigidity). Some first-order motions are called trivial because they exist irrespective of the details of the framework: infinitesimal translations and rotations. They can be written in the form \\(\\dot {\\boldsymbol p} = S\\boldsymbol p + v\\) where \\(S\\in\\Bbb R^{d\\times d}\\) is skew-symmetric and \\(v\\in\\Bbb R^d\\). A first-order motion that is not trivial is called a first-order flex. A framework that has a first-order flex is said to be first-order flexible. It is called first-order rigid if it has none. First-order rigidity implies actual rigidity, whereas first-order flexibility does not always imply actual flexibility.\nFor the linear system \\((*)\\) we can write down a matrix \\(R(G,\\boldsymbol p)\\in\\Bbb R^{E\\times dV}\\) whose rows correspond to the individual equations. Then \\(\\dot{\\boldsymbol p}\\) is a first-order motion if and only if \\(R(G,\\boldsymbol p)\\dot{\\boldsymbol p}=0\\), or equivalenty, if \\(\\dot{\\boldsymbol p}\\in\\ker R(G,\\boldsymbol p)\\). This matrix is called the rigidity matrix of the framework. It has the following structure:\n\nThe rigidity matrix is a useful way of packaging the linear details of a framework. Essentially all of first-order theory can be expressed rather elegantly using it. For example, a framework (with \\(V\\ge d+1\\) vertices) is first-order rigid if and only if \\(\\operatorname{corank}R(G,\\boldsymbol p)={d+1\\choose 2}\\). Moreover, the rank of the rigidity matrix can be seen as a more fine-grained measure of rigidity, or of “how far away from rigid” a framework is (in the first-order sense).\nThe other first-order notion that is often useful is a stress, which is a map \\(\\boldsymbol\\omega:E\\to\\Bbb R\\).  An equilibrium stress is a stress that satisfies\n\\[(**)\\quad \\sum_{\\mathclap{j:ij\\in E}}\\omega_{ij}(p_j-p_i)=0,\\quad\\text{for all $i\\in V$}.\\]\nThe physical interpretation of stress is as an internal force in an edge that tries to expand or contract the edge by pushing and pulling on its end vertices along the edge direction. An equilibrium stress is then a stress where at each vertex these forces cancel out. It turns out that \\(\\boldsymbol\\omega\\) is an equilibrium stress if and only if \\(R^\\top\\!(G,\\boldsymbol p)\\boldsymbol\\omega=0\\), or equivalently, if \\(\\boldsymbol\\omega\\in\\ker R^\\top\\!(G,\\boldsymbol p)=\\mathop{\\mathrm{coker}}R(G,\\boldsymbol p)\\).\n\nSince first-order motions and stresses appear as the kernel and cokernel of the rigidity matrix respectively, one can use either to study its rank, and hence, the first-order rigidity of \\((G,\\boldsymbol p)\\). In fact, even though one usually aims to make a statement about first-order flexes, it often turns out more convenient to argue about stresses. One reason is that stresses have no equivalent of “trivial motions” one needs to account for. We will see an example of this when we consider coned frameworks."
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    "title": "Coned frameworks and radial moves",
    "section": "Coned frameworks and radial moves",
    "text": "Coned frameworks and radial moves\nA framework \\((G,\\boldsymbol p)\\) is coned if the graph \\(G\\) has a dominating vertex, that is, a vertex that is adjacent to all other vertices. We shall denote this cone vertex by \\(*\\in V\\).\nA radial move in a coned framework slides vertices (other than the cone vertex) along the rays that connect them to the cone vertex. If we assume for simplicity that \\(p_*=0\\), then a radial move is given by \\(p_i\\mapsto r_i p_i\\) for \\(r_i\\not=0\\).\n\n\nTheorem   Radial moves preserve first-order rigidity and flexibility. More strongly, radial moves preserve the rank of the rigidity matrix. \n\nProof.\nLet \\((G,\\boldsymbol p)\\) be a coned framework, and \\((G,\\hat{\\boldsymbol p})\\) obtained from it by radial moves. As before, we assum \\(p_*=0\\) and \\(\\hat p_i=r_i p_i\\) for \\(r_i\\not=0\\).\nWith the primary goal to show that radial moves preserve first-order rigidity, which is defined in terms of first-order flexes, ones first attempt is perhaps to construct an explicit one-to-one correspondence between first-order flexes of \\((G,\\boldsymbol p)\\) and \\((G,\\hat{\\boldsymbol p})\\). This turns out quite tedious. Instead, following a previous comment, the trick is to construct a one-to-one correspondence between equilibrium stresses. Since\n\\[\\operatorname{rank}R(G,\\boldsymbol p)=\\operatorname{rank}R(G,\\boldsymbol p)^\\top=E-\\!\\overbrace{\\,\\ker R(G,\\boldsymbol p)^\\top\\,}^{\\mathclap{\\text{space of equilibrium stresses}}}\\!,\\]\nthis also shows that the rank of the rigidity matrix is preserved.\n\nSuppose then that we are given an (equilibrium) stress \\(\\boldsymbol\\omega:E\\to\\Bbb R\\) of \\((G,\\boldsymbol p)\\), and let us try to construct an equilibrium stress \\(\\hat{\\boldsymbol\\omega}:E\\to\\Bbb R\\) for the radially rescaled framework \\((G,\\hat{\\boldsymbol p})\\), so as to obtain a one-to-one map between the stress spaces. We shall adopt the convention that \\(\\omega_{ij}=0\\) whenever \\(ij\\not\\in E\\) or \\(i=j\\), which makes many expressions less cluttered.\nWe make the natural guess \\(\\hat\\omega_{ij}:=\\omega_{ij}/r_ir_j\\) for the non-cone edges \\(ij\\in E\\). The corresponding natural guess \\(\\hat\\omega_{i*}:=\\omega_{i*}/r_i^2\\) for cone edges does actually not work, and the correct choice is hardly guessable. Instead we derive it by enforcing the equilibrium condition \\((**)\\) at vertex \\(i\\not=*\\):\n\\[\\begin{align}\n    0\\;\n    &\\overset!=\n    \\sum_{j} \\hat\\omega_{ij} (\\hat p_j-\\hat p_i)\n    \\\\&=\n    \\sum_{\\mathclap{j\\not=*}} \\hat\\omega_{ij} (\\hat p_j-\\hat p_i) + \\smash{\\overbrace{\\hat\\omega_{i*} (\\hat p_*-\\hat p_i)}^{-\\hat\\omega_{i*}\\hat p_i\\mathrlap{\\text{\\color{lightgray}$\\;\\;\\leftarrow$ we used $p_*=0$}}}}\n    \\\\&=\n    \\sum_{\\mathclap{j\\not=*}} \\frac{\\omega_{ij}}{r_i r_j} (r_jp_j- r_ip_i) -\\hat\\omega_{i*}r_ip_i\n    \\\\&=\n    \\tfrac1{r_i} \\sum_{\\mathclap{j\\not=*}} \\omega_{ij} p_j + \\Big(\\sum_{\\mathclap{j\\not=*}} \\omega_{ij} \\tfrac1{r_j} \\Big)p_i -\\hat\\omega_{i*}r_ip_i\n    \\\\&=\n    \\tfrac1{r_i} \\underbrace{\\sum_{\\mathclap{j\\not=*}} \\omega_{ij} (p_j - p_i)}_{-\\omega_{i*}(p_*-p_i)\\,=\\, \\omega_{i*}p_i \\mathrlap{\\text{\\color{lightgray}$\\;\\;\\leftarrow$ we used $(**)$ for $\\boldsymbol\\omega$ at vertex $i$}}} +\n    \\Bigg(\\sum_{\\mathclap{j\\not=*}} \\omega_{ij} \\Big(\\tfrac1{r_j}-\\tfrac1{r_i}\\Big) - \\hat\\omega_{i*}r_i \\Bigg)p_i\n    \\\\&=\n    \\Bigg(\\tfrac1{r_i}\\omega_{i*} + \\sum_{\\mathclap{j\\not=*}} \\omega_{ij} \\Big(\\tfrac1{r_j}-\\tfrac1{r_i}\\Big) - \\hat\\omega_{i*}r_i\\Bigg)p_i\n\\end{align}\\]\nThus, if \\(p_i\\not= 0\\), we necessarily have to set\n\\[\n\\hat\\omega_{i*}:= \\frac{\\omega_{i*}}{r_i^2} + \\frac{1}{r_i}\\sum_{j\\not=*} \\omega_{ij} \\Big(\\tfrac1{r_i}-\\tfrac1{r_j}\\Big).\n\\]\n(and if \\(p_i=0\\), then the we can set \\(\\hat\\omega_{i*}\\) arbitrarily). This is not far from our ansatz: it is the natural guess \\(\\omega_{i*}/r_i^2\\) plus some reasonably nice correction term.\nIt remains to check that \\(\\hat{\\boldsymbol\\omega}\\) satisfies the equilibrium condition \\((**)\\) also at the cone vertex. But this can be inferred without much computation. For the students in my course I argue using some terminology (for everyone else, see the self-contained argument below): the stress \\(\\hat{\\boldsymbol\\omega}\\) gives rise to a resolvable load \\(f_i:=-\\sum_j \\hat\\omega_{ij}(\\hat p_j-\\hat p_i)\\). As shown above, \\(f_i=0\\) for all \\(i\\not=*\\). But resolvable loads are equilibrium loads, in particular, \\(\\sum f_i=0\\). Thus, \\(f_*=0\\) follows as well.\nThis can be argued without the slang: if a stress \\(\\hat{\\boldsymbol\\omega}\\) satisfies the equilibrium condition \\((**)\\) at all vertices \\(i\\not=*\\), then it can be inferred to hold at \\(i=*\\) as well: \n\\[\\begin{align}\n\\sum_i \\hat\\omega_{i*} (\\hat p_*-\\hat p_i)\n&= \\sum_{i,j} \\hat \\omega_{ij} (\\hat p_j - \\hat p_i) - \\sum_{j\\not=*}\\overbrace{\\sum_{i} \\hat \\omega_{ij} (\\hat p_j - \\hat p_i)}^{=0}\n\\\\&= \\sum_{i<j} \\hat \\omega_{ij} (\\hat p_j-\\hat p_i) + \\sum_{i>j}\\hat \\omega_{ij}(\\hat p_j-\\hat p_i)\n\\\\&= \\sum_{i<j} \\hat \\omega_{ij} (\\hat p_j-\\hat p_i) + \\sum_{j>i}\\hat \\omega_{ji}\\smash{\\overset{\\mathrlap{\\;\\text{\\color{lightgray}(we renamed $i$ and $j$)}}}(}\\hat p_i-\\hat p_j)\n\\\\&= \\sum_{i<j} \\hat \\omega_{ij} (\\hat p_j-\\hat p_i) - \\sum_{i < j}\\hat \\omega_{ij}(\\hat p_j-\\hat p_i)\n=0\n\\end{align}\\]\nWe have conclusively shown that \\(\\hat{\\boldsymbol\\omega}\\) is an equilibrium stress for \\((G,\\hat{\\boldsymbol p})\\). Also, \\(\\hat{\\boldsymbol\\omega}\\) is a linear expression in terms of \\(\\boldsymbol\\omega\\), and clearly an invertible one (use \\(1/r_i\\) instead of \\(r_i\\) for the inverse). Hence we described a linear isomorphism between the equilibrium stress spaces of \\((G,\\boldsymbol p)\\) and \\((G,\\hat{\\boldsymbol p})\\). This concludes the proof.\n\n\\(\\square\\)"
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