Proofs on Ball-Polyhedra and Spindle Convex Bodies

Abstract

First, we prove that any of the shortest generalized billiard trajectories in an arbitrary convex body of Euclidean d-space is of period of at most d + 1. Second, we prove an analogue of Stoker’s rigidity theorem for standard ball-polyhedra. Third, we give a proof of the global rigidity analogue of Alexandrov’s theorem for normal ball-polyhedra. Next, we show that every simple and standard ball-polyhedron of Euclidean 3-space is locally rigid with respect to its inner dihedral angles (resp., face angles). Then we prove some basic separation and support properties for spindle convex bodies as well as give a proof of a Charathéodory-type theorem for spindle convex hulls. Furthermore, we prove an Euler-Poincaré-type formula for standard ball-polyhedra in Euclidean d-space. Finally, we give a proof of the long-standing Boltyanski-Hadwiger illumination conjecture for fat spindle convex bodies in Euclidean dimensions greater than or equal to 15.KeywordsDihedral AngleConvex BodyVoronoi CellConstant WidthConvex PolyhedronThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.