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

Lizhao Zhang

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arXiv · 2307.15817

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

Lizhao Zhang

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Abstract

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 $p\ge 2$.

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BibTeX RIS

Lizhao Zhang. 2025-01-20. Simplices with fixed volumes of codimension 2 faces in a continuous deformation. https://arxiv.org/abs/2307.15817

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