Discrete isoperimetric problems in spaces of constant curvature

TL;DR

This paper establishes isoperimetric inequalities for simplices and polytopes with d+2 vertices across Euclidean, spherical, and hyperbolic spaces, identifying minimal and maximal volume configurations under various constraints.

Contribution

It introduces new isoperimetric inequalities for polytopes with d+2 vertices in different constant curvature spaces, using Steiner symmetrization as a key tool.

Findings

Minimal volume hyperbolic simplices with given inradius identified

Maximal volume spherical polytopes with fixed circumradius characterized

Polytopes with minimal total edge length and fixed inradius analyzed

Abstract

The aim of this paper is to prove isoperimetric inequalities for simplices and polytopes with $d+2$ vertices in Euclidean, spherical and hyperbolic $d$-space. In particular, we find the minimal volume $d$-dimensional hyperbolic simplices and spherical tetrahedra of a given inradius. Furthermore, we investigate the properties of maximal volume spherical and hyperbolic polytopes with $d+2$ vertices with a given circumradius, and the hyperbolic polytopes with $d+2$ vertices with a given inradius and having a minimal volume or minimal total edge length. Finally, for any $1 \leq k \leq d$, we investigate the properties of Euclidean simplices and polytopes with $d+2$ vertices having a fixed inradius and a minimal volume of its $k$-skeleton. The main tool of our investigation is Euclidean, spherical and hyperbolic Steiner symmetrization.