Ball-Polyhedra as Intersections of Congruent Balls

Abstract

The previous sections indicate a good deal of geometry on unions and intersections of balls that is worthwhile studying. In particular, when we restrict our attention to intersections of balls the underlying convexity suggests a broad spectrum of new analytic and combinatorial results. To make the setup ideal for discrete geometry from now on we look at intersections of finitely many congruent closed d-dimensional balls with non-empty interior in $\mathbb{E}^d$. In fact, one may assume that the congruent d-dimensional balls in question are of unit radius; that is, they are unit balls of $\mathbb{E}^d$. Also, it is natural to assume that removing any of the unit balls defining the intersection in question yields the intersection of the remaining unit balls becoming a larger set. Id d = 2, then we call the sets in question disk-polygons and for d ≥ 3 ther are called ball-polyhedra. This definition along with some basic properties of ball-polyhedra (resp., disk-polygons) were introduced by the author in a sequence of talks at the University of Calgary in the fall of 2004. Based on that, the paper [69] written by the author, Lángi, Naszódi, and Papez systematically extended those investigations to get a better understanding of the geometry of ballpolyhedra (resp., disk-polygons) by proving a number of theorems, which one can regard as the analogues of the classical theorems on convex polytopes.