Description
We examine several metric and combinatorial properties of ball-bodies and ball-polyhedra.
First, we establish Blaschke–Santaló-type inequalities for r-ball bodies. These results allow us to extend earlier work on analogues of the Kneser-Poulsen conjecture, specifically for intersections of balls under uniform contractions in Euclidean d-space. As a direct consequence, we obtain a proof of Alexander’s conjecture in the setting of uniform contractions.
We then introduce the class of basic r-ball polyhedra in Euclidean d-space and analyze their face structure. In this context, we prove an analogue of McMullen’s Upper Bound Theorem. Finally, we show that every basic r-ball polyhedron is globally rigid with respect to its inner dihedral angles.
The talk will also be broadcast on Zoom.