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$d+2$ 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$d+2$ vertices with a given circumradius, and the hyperbolic polytopes with d+2$d+2$ vertices with a given inradius and having a minimal volume or minimal total edge length. Finally, for any 1⩽k⩽d$1 \leqslant k \leqslant d$, we investigate the properties of Euclidean simplices and polytopes with d+2$d+2$ 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.