Some geometric properties of regular polyhedral complexes

Abstract

In the previous paper [2], we have classified simple regular polyhedral BP-complexes, which are polyhedral complexes satisfying certain natural conditions (B) and (P) on their vertex structures. A regular polyhedral complex X is defined as satisfying the condition (B) if the diameter of the vertex structure of X is equal to its injectivity radius. On the other hand, we have shown in [3] that 2-skeletons of higher dimensional regular polytopes do not satisfy the condition (B). So it seems very likely that the diameter of the vertex structure and its injective radius are not equal.

In this short paper, we consider the geometric implications of the diameter and the injectivity radius of the vertex structure of regular polyhedral complexes.