Joint Separation of Geometric Clusters and the Extreme Irregularities of Regular Polyhedra

Abstract

We propose a natural scheme to measure the (so-called) joint separation of a cluster of objects in general geometric settings. In particular, here the measure is developed for finite sets of planes in ℝ³ in terms of extreme configurations of vectors on the planes of a given set. We prove geometric and graph-theoretic results about extreme configurations on arbitrary finite plane sets. We then specialize to the planes bounding a regular polyhedron in order to exploit the symmetries. However, even then results are non-trivial and surprising – extreme configurations on regular polyhedra may turn out to be highly irregular.