Universality theorems for inscribed polytopes and Delaunay triangulations

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. ...

Many neighborly inscribed polytopes and Delaunay triangulations

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, a...

Lectures on Discrete and Polyhedral Geometry

1. The Helly theorem 11 2. Carathéodory and Bárány theorems 20 3. The Borsuk conjecture 26 4. Fair division 32 5. Inscribed and circumscribed polygons 39 6. Dyson and Kakutani theorems 54 7. Geometric inequalities 61 8. Combinatorics of convex polytopes 72 9. Center of mass, billiards and the variational principle 83 10. Geodesics and quasi-geodesics 96 11. The Steinitz theorem and its extensio...

Inscribable polytopes via Delaunay triangulations

ay 2 00 8 Notes on Convex Sets , Polytopes , Polyhedra Combinatorial Topology , Voronoi Diagrams and Delaunay Triangulations

Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended f...