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 the talk we survey the state of the art of the Kneser-Poulsen conjecture. Based on that it seems very natural and important to study the geometry of intersections of finitely many congruent balls say, of unit balls, from the viewpoint of discrete geometry in Euclidean space. We call these sets ball-polyhedra. In the second half of the talk we survey a selection of fundamental results known on ball-polyhedra.