Ball-polyhedra (Q2464357)

The authors study geometric and further properties of intersections of finitely many congruent balls (called ball-polyhedra) in \(\mathbb{R}^n\) and \(\mathbb{R}^3\), mainly from the viewpoint of discrete and combinatorial geometry. They also discuss the relation of ball-polyhedra to notions like spindle convexity and Reuleaux polytopes. Important topics presented in this (partially also expository) paper are: ball-polyhedra and famous theorems from combinatorial geometry (such as statements related to Kirchberger's and Carathéodory's theorem), results related to the Euler-Poincaré formula or extending Johnson's three-circles theorem, basic observations referring to the inradius, thickness, the edge-graph, and illumination of ball-polyhedra, Dowker-type isoperimetric inequalities and Erdős-Szekeres-type problems. In most cases, the authors give starting points for further research on ball-polyhedra.