Rigidity and volume preserving deformation on degenerate simplices

Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\geq n$), for each $k$, $1\leq k\leq n$, 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 $d\geq n+1$ 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.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr

Simplices with fixed volumes of codimension 2 faces in a continuous deformation

For any $n$-dimensional simplex in the Euclidean space $\mathbb{R}^n$ with $n\ge 4$, it is asked that if a continuous deformation preserves the volumes of all the codimension 2 faces, then is it necessarily a \emph{rigid} motion. While the question remains open and the general belief is that the answer is affirmative, for all $n\ge 4$, we provide counterexamples to a variant of the question where $\mathbb{R}^n$ is replaced by a pseudo-Euclidean space $\mathbb{R}^{p,n-p}$ for some unspecified $p\ge 2$.Comment: 16 pages, 1 figure. major revision, changed the title, added counterexample for $n=4$ in $\mathbb{R}^{3,1}$ as well, so we now have counterexamples for all $n\ge 4