• 1 Institut für Mathematik

We present a very simple explicit technique to generate a large family of point configurations with neighborly Delaunay triangulations. This proves that there are superexponentially many combinatorially distinct neighborly $d$-polytopes with $n$ vertices that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (and thus also of Delaunay triangulations). It coincides with the current best lower bound for the number of combinatorial types of polytopes.