From the Kneser–Poulsen conjecture to ball-polyhedra

A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass, Arch. Math. 6 (1955) 382–390] and Poulsen E.T. Poulsen, Problem 10, Math. Scand. 2 (1954) 346]. 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.