From the Kneser-Poulsen conjecture to ball-polyhedra

TL;DR

This paper surveys the Kneser-Poulsen conjecture across different geometries and explores the geometry of ball-polyhedra, aiming to motivate further research with open problems.

Contribution

It provides a comprehensive survey of the Kneser-Poulsen conjecture and introduces the study of ball-polyhedra in Euclidean space, highlighting open problems for future work.

Findings

Survey of the Kneser-Poulsen conjecture in Euclidean, spherical, and hyperbolic spaces

Introduction and analysis of the geometry of ball-polyhedra in Euclidean space

Identification of open problems to guide future research

Abstract

A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser (1955) and Poulsen (1954). According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that the distance between the centers of every pair of balls is decreased, then the volume of the union (resp., intersection) of the balls is decreased (resp., increased). In the first half of this paper we survey the state of the art of the Kneser-Poulsen conjecture in Euclidean, spherical as well as hyperbolic spaces with the emphases being on the Euclidean case. Based on that it seems very natural and important to study the geometry of intersections of finitely many congruent balls from the viewpoint of discrete geometry in Euclidean space. We call these sets ball-polyhedra. In the second half of this paper we survey a selection of fundamental results known on ball-polyhedra. Besides the obvious survey character of this paper we want to emphasize our definite intention to raise quite a number of open problems to motivate further research.