Delaunay polytopes of cut lattices

Delaunay polytopes of cut lattices

We continue 3], the study of the lattice L n generated by cuts of the complete graph on a set V n of n vertices. The lattice L n spans an N = ? n 2-dimensional space of all functions deened on a set V 2 n of all unordered pairs of the set V n. We prove that the cut polytope, i.e. the convex hull of all cuts, is an asymmetric Delaunay polytope of L n. Symmetric Delaunay polytopes of a lattice L ...

Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices

A polytope D, whose vertices belong to a lattice of rank d, is Delaunay if it can be circumscribed by an ellipsoid E with interior free of lattice points, and so that the vertices of D are the only lattice points on the quadratic surface E. If in addition E is uniquely determined by D, we call D a perfect Delaunay polytope. Thus, in the perfect case, the lattice points on E, which are the verti...

Inscribable polytopes via Delaunay triangulations

The six-dimensional Delaunay polytopes

Given a lattice L, a full dimensional polytope P is called a Delaunay polytope if the set of its vertices is S ∩ L with S being an empty sphere of the lattice. Extending our previous work [DD01] on the hypermetric cone HY P7, we classify the six-dimensional Delaunay polytopes according to their combinatorial type. The list of 6241 combinatorial types is obtained by a study of the set of faces o...

Neighborly inscribed polytopes and Delaunay triangulations

We prove that there are superexponentially many combinatorially distinct d-dimensional neighborly Delaunay triangulations on n points. These are the first examples of neighborly Delaunay triangulations that cannot be obtained via a stereographic projection of an inscribed cyclic polytope, and provide the current best lower bound for the number of combinatorial types of Delaunay triangulations. ...