The infimum of the volumes of convex polytopes of any given facet areas is 0

TL;DR

This paper proves that in Euclidean space of dimension three or higher, the infimum of the volumes of convex polytopes with specified facet areas is zero, extending previous results and exploring spherical and hyperbolic geometries.

Contribution

It establishes the zero infimum of volumes for convex polytopes with given facet areas in higher-dimensional Euclidean spaces and provides necessary conditions in spherical and hyperbolic geometries.

Findings

Infimum of volumes in Euclidean space is zero for given facet areas.

Necessary conditions for convex polytopes in spherical and hyperbolic spaces.

Partial results on existence of convex tetrahedra with prescribed facet areas.

Abstract

We prove the theorem mentioned in the title, for ${\mathbb{R}}^n$, where $n \ge 3$. The case of the simplex was known previously. Also, the case $n=2$ was settled, but there the infimum was some well-defined function of the side lengths. We also consider the cases of spherical and hyperbolic $n$-spaces. There we give some necessary conditions for the existence of a convex polytope with given facet areas, and some partial results about sufficient conditions for the existence of (convex) tetrahedra.