Normally inscribable polytopes

A polytope $P$ is inscribable if there is a combinatorially equivalent polytope $P'$ with all vertices contained in a sphere. This notion relates to the combinatorics of ideal hyperbolic polytopes and Delaunay subdivisions. Steinitz showed that not every polytope is inscribable and Rivin gave a complete and effective characterization for $3$-dimensional polytopes. There is no such characterization in dimensions $4$ and up.

I will discuss the related notion of normally inscribable polytopes: A polytope $P$ is normally inscribable if there is a normally equivalent polytope $P'$ with all vertices on a sphere. Normal inscribability, it turns out, reveals a fascinating interplay of algebra, geometry, and combinatorics. I will explain connections to a deformation theory of ideal hyperbolic polytopes and Delaunay subdivisions, routed particle trajectories, and reflection groupoids. In particular, for zonotopes, this reveals new connections to reflection groups and Grünbaum's quest for simplicial arrangements.

This is based on joint work with Sebastian Manecke.