Analogues of Alexandrov's and Stoker's theorems for ball-polyhedra

TL;DR

This paper extends classical rigidity theorems from convex polyhedra to ball-polyhedra, which are intersections of congruent balls, providing new insights into their geometric properties.

Contribution

It establishes analogues of Alexandrov's and Stoker's rigidity theorems specifically for normal and standard ball-polyhedra in three-dimensional space.

Findings

Proves rigidity properties for ball-polyhedra

Extends classical theorems to non-convex intersections

Provides a foundation for further geometric analysis of ball-polyhedra

Abstract

The rigidity theorems of Alexandrov (1950) and Stoker (1968) are classical results in the theory of convex polyhedra. In this paper we prove analogues of them for normal (resp., standard) ball-polyhedra. Here, a ball-polyhedron means an intersection of finitely many congruent balls in Euclidean 3-space.