Simplices of maximal volume or minimal total edge length in hyperbolic space.

Abstract

This article is mainly concerned with simplices in n-dimensional hyperbolic space. The main tool is a hyperbolic version of Steiner symmetrization. Our main results are: (A) Let T be the set of all hyperbolic n-simplices in a given closed ball B. A simplex in T is of maximal volume if and only if it is regular and if its vertices are contained in the boundary of B. (B) A hyperbolic simplex is of maximal volume if and only if it is regular and ideal. (C) Let T denote the set of all finite hyperbolic simplices with inradius r. A simplex in T has minimal total edge length if and only if it is regular. (D) Let T denote the set of all finite hyperbolic simplices of volume V. A simplex in T has minimal total edge length if and only if it is regular.