Rigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints

Abstract

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 $P\subset\mathbb R^d$ and $Q\subset\mathbb R^e$ 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 $q\colon V(G_P)\to\mathbb R^e$ of the edge-graph $G_P$ 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.