Large Structures Seminar: Arnau Padrol

Delaunay triangulations are a central object in many areas of mathematics and computational geometry. However, little is known about which combinatorial types can appear as a Delaunay triangulation. With an inverse stereographic projection, this question translates to asking which combinatorial types of polytopes admit realizations with all the vertices on a sphere. We will construct a large family of neighborly polytopes that are not only inscribable on the sphere, but also in every smooth strictly convex body. This provides the current best lower bound for the number of combinatorial types of Delaunay triangulations. In the second part of the talk, we will use these techniques to show that realization spaces of inscribed polytopes present Universality in the sense of Mnëv. For example, this will allow us to construct a pair of point configurations whose Delaunay triangulations are combinatorially equivalent but yet there is no continuous transformation that maps one to the other without changing the Delaunay triangulation.

This talk reports joint work with Karim Adiprasito, Bernd Gonska and Louis Theran.