Rigid Ball-Polyhedra in Euclidean 3-Space

A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ballpolyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ball-polyhedron if its vertex-edge-face structure is a lattice (with respect to containment). To each edge of a ball-polyhedron one can assign an inner dihedral angle and say that the given ball-polyhedron is locally rigid with respect to its inner dihedral angles if the vertexedge-face structure of the ball-polyhedron and its inner dihedral angles determine the ball-polyhedron up to congruence locally. The main result of this paper is a Cauchytype rigidity theorem for ball-polyhedra stating that any simple and standard ballpolyhedron is locally rigid with respect to its inner dihedral angles.