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 Carathéodory-type theorem for spindle convex hulls, and an Euler–Poincaré-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.
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Bezdek, K. (2013). Ball-Polyhedra and Spindle Convex Bodies. In: Lectures on Sphere Arrangements – the Discrete Geometric Side. Fields Institute Monographs, vol 32. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8118-8_5
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