Ball-Polyhedra and Spindle Convex Bodies

Abstract

In this chapter, we introduce an extension of the theory of convex polyhedral sets (resp., convex bodies) to the family of intersections of finitely many congruent balls, called ball-polyhedra (resp., to the family of intersections of not necessarily finitely many congruent balls, called spindle convex bodies). The basic properties of ball-polyhedra (resp., spindle convex bodies) that are discussed here include separation and support properties for spindle convex bodies, a Caratheodory-type theorem for spindle convex hulls, and an Euler–Poincare-type formula for ball-polyhedra in d-dimensional Euclidean space. In addition, we discuss (generalized) billiard trajectories in disk-polygons and an analogue of Blaschke–Lebesgue theorem for disk-polygons. Furthermore, we investigate the problem of characterizing the edge-graphs of ball-polyhedra in Euclidean 3-space. Another topic, discussed in more details, is on global and local rigidity of ball-polyhedra in Euclidean 3-space. Finally, we investigate the status of the long-standing illumination conjecture of V. G. Boltyanski and H. Hadwiger from 1960 for ball-polyhedra (resp., spindle convex bodies) in Euclidean d-space.