Discrete isoperimetric problems in spaces of constant curvature

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 ≤ k ≤ 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.