Ball-polyhedra and antipodality

Abstract

We investigate antipodality and ball-polyhedra. We call a set A in Euclidean d-space antipodal if, through any two points of A, there is a pair of parallel hyperplanes that pass through those points and support A. First, as an application of antipodality, we prove an upper bound on the cardinal­ity of pairwise touching, positive homothetic, copies of a convex body in Euclidean d-space. The problem of finding a tight upper bound was posed by K. Bezdek and J. Pach in 1988. We extend the definition of antipodality to Hyperbolic d-space in three different ways, prove upper bounds for the maximum cardinality of sets that satisfy these definitions and show connections to Euclidean antipodality. Then, we turn our attention to intersections of finitely many balls of the same radius in Euclidean d-space. These objects are called ball-polyhedra. As a starting point, we lay the foundations of the study of ball-polyhedra, and a notion that appears naturally, spindle convexity. Next, we investigate analogues of theorems from the theory of linear convexity and (linear) polyhedral sets. Then, we prove the following isoperimetric result in the Euclidean, Hyperbolic and Spherical Planes: Among all "n-gons" with unit circular arc edges and with a given perimeter, the regular one encloses the largest area. Next, illumination, the face lattice and a symmetry property of three-dimensional ball-polyhedra are discussed. Finally, we study rigidity of ball-polyhedra; that is, the problem of finding parameters that determine a ball-polyhedron uniquely.