2023/02/27 by Martin Winter, Winter, Martin · 2 citations
Computer Science · Engineering · Mathematics · #Computational Geometry and Mesh Generation #Structural Analysis and Optimization #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2302.14194
We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types. We conjecture even stronger that for two polytopes P⊂\mathbb Rd and Q⊂\mathbb Re with the same edge-graph it is not possible that Q has longer edges than P while also having smaller vertex-point distances. We develop techniques to attack this question and verify it in three relevant special cases: if P and Q are centrally symmetric, if Q is a slight perturbation of P, and if P and Q are combinatorially equivalent. In the first two cases the statements stay true if we replace Q by some graph embedding q\colon V(GP)→\mathbb Re of the edge-graph GP of P, which can be interpreted as local resp. universal rigidity of certain tensegrity frameworks. We also establish that a polytope is uniquely determined up to affine equivalence by its edge-graph, edge lengths and the Wachspress coordinates of an arbitrary interior point. We close with a broad overview of related and subsequent questions.