Rigidity and volume preserving deformation on degenerate simplices

Abstract

Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\ge n$), for each $k$, $1\le k\le n$, Radon's theorem induces a partition of the set of $k$-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in $M^d$ for $d=n$, and the volumes of $k$-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all $k$-faces; and this property still holds in $M^d$ for $d\ge n+1$ if an invariant $c_{k-1}({\alpha }^{k-1})$ of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant $c_k(\omega )$ we discovered for any $k$-stress $\omega$ on a cell complex in $M^d$. We introduce a characteristic polynomial of the degenerate simplex by defining $f(x)={\sum }_{i=0}^{n+1}(-1{)}^i{c}_i({\alpha }^i)x^{n+1-i}$, and prove that the roots of $f(x)$ are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case.

27 pages, 2 figures. To appear in Discrete & Computational Geometry

Cited by in corpus (2)

• Lifting degenerate simplices with a single volume constraint
• On the total volume of the double hyperbolic space