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. © Springer-Verlag Berlin Heidelberg 2003.