Bipartite Sets of Spheres and Casey-Type Theorems

Abstract

A bipartite set of spheres in ${\mathbb {R}}^n$ is a set of colored spheres, where two colors are used, no sphere is contained in the closed ball bounded by another sphere in the set, and spheres of different colors are disjoint. For any two spheres in a bipartite set, the common-tangent distance between them is defined as the distance between two tangent points in a common tangent hyperplane to them, where an external common tangent hyperplane is taken if the two spheres are of the same color; otherwise, a common internal tangent hyperplane is taken. By this common-tangent distance, a bipartite set becomes a semi-metric space. It turns out that bipartite sets of spheres form an interesting family of semi-metric spaces. Casey’s theorem (a generalization of Ptolemy’s theorem) gives a condition for a bipartite set of four circles in ${\mathbb {R}}^2$ to have a circle that is suitably tangent to all circles in the bipartite set. Ptolemy’s theorem is generalized to the n-dimensional situation via Cayley–Menger determinants. Among other results, we present a Casey-type theorem for a bipartite set of $n+2$ spheres in n dimensions as a generalization of the n-dimensional version of Ptolemy’s theorem, and we extend this further to a bipartite set with an arbitrary number of spheres in ${\mathbb {R}}^n$.