Simplicial subdivision of simplices of arbitrary dimension in spaces of constant curvature with bounded quality

Abstract

In 1942, Freudenthal showed that a simplex in Euclidean space can be subdivided such that the quality (well-shapedness of the simplex, quantified in terms of e.g. fatness) of the simplices in the subdivision is lower bounded. This answered a question of Brouwer. Recently, Brunck discussed the same problem for simplices in two-dimensional spaces of constant curvature and provided a closely related construction. In this paper we generalize Brunck's result to arbitrary dimensional spaces of constant curvature by combining Freudenthal's construction and radial projection. We contrast this approach with Brunck's construction.