Ball-Polyhedra

TL;DR

This paper explores properties of ball-polyhedra, which are intersections of finitely many equal-radius balls, and introduces spindle convexity, providing analogues of classical convex polyhedral results.

Contribution

It introduces and analyzes the concept of spindle convexity and extends classical convex polyhedral results to the setting of ball-polyhedra.

Findings

Ball-polyhedra are intersections of finitely many equal-radius balls.

Analogues of convex polyhedral set results are established for ball-polyhedra.

Spindle convexity is characterized and related to ball-polyhedra.

Abstract

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.