On polytopes with congruent projections or sections

TL;DR

This paper proves that convex polytopes with congruent projections or sections onto all k-dimensional subspaces are necessarily translates or reflections of each other, revealing a strong geometric uniqueness property.

Contribution

It establishes that polytopes with congruent projections or sections onto all k-dimensional subspaces are related by translation or reflection, extending known geometric rigidity results.

Findings

Polytopes with congruent projections are translates or reflections.

Polytopes with congruent sections are identical or reflections, given interior point condition.

Results hold for all k-dimensional subspaces with 2 ≤ k ≤ d-1.

Abstract

Let $2\le k\le d-1$ and let $P$ and $Q$ be two convex polytopes in ${\mathbb E^d}$. Assume that their projections, $P|H$, $Q|H$, onto every $k$-dimensional subspace $H$, are congruent. In this paper we show that $P$ and $Q$ or $P$ and $-Q$ are translates of each other. We also prove an analogous result for sections by showing that $P=Q$ or $P=-Q$, provided the polytopes contain the origin in their interior and their sections, $P \cap H$, $Q \cap H$, by every $k$-dimensional subspace $H$, are congruent.