Selected topics from the theory of intersections of balls

TL;DR

This survey explores volumetric and combinatorial properties of intersections and unions of balls in Euclidean space, discussing key conjectures, inequalities, and structures like ball-polyhedra with new insights into their geometry.

Contribution

It provides a comprehensive overview of recent results and open problems related to intersections and unions of balls, including the Kneser--Poulsen Conjecture and properties of ball-polyhedra.

Findings

Kneser--Poulsen Conjecture relates volume changes to rearrangements of balls.

New inequalities for $r$-ball bodies and their geometric properties.

Analysis of the combinatorial structure and volumetric properties of ball-polyhedra.

Abstract

In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the $d$-dimensional real vector space, mostly equipped with the Euclidean norm. Our first topic is the Kneser--Poulsen Conjecture, according to which if a finite number of balls are rearranged so that the pairwise distances of the centers increase, then the volume of the union (resp., intersection) increases (resp., decreases). Next, we discuss Blaschke--Santal\'o-type inequalities, and reverse isoperimetric inequalities for convex sets in Euclidean $d$-space obtained as intersections of (possibly infinitely many) balls of radius $r$, which we call $r$-ball bodies. We present some results on $1$-ball bodies (also called ball-bodies or spindle convex sets) in the plane, with special attention paid to their approximation by the spindle convex hull of a finite subset. A ball-polyhedron is a ball-body obtained as the intersection of finitely many unit balls in Euclidean $d$-space. We consider the combinatorial structure of their faces, and volumetric properties of ball-polyhedra obtained from choosing the centers of the balls randomly.