Classical Topics in Discrete Geometry pp 57-68 | Cite as

# 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.

## Keywords

Dihedral Angle Unit Ball Convex Body Convex Domain Face Lattice## Preview

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