Neighborly inscribed polytopes and delaunay triangulations

Abstract

Abstract We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that for d ≥ 4 there are superexponentially many combinatorially distinct neighborly d-polytopes on n vertices that admit realizations inscribed in 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 (which coincides with the current best lower bound for the number of combinatorial types of polytopes). Via stereographic projections, this translates into a superexponential lower bound for the number of combinatorial types of (neighborly) Delaunay triangulations in ℝ d for d ≥ 3.