Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations

TL;DR

This paper provides a comprehensive tutorial on convex sets, polytopes, polyhedra, and related topological and geometric concepts, including new insights into projective polyhedra and proofs of polyhedron equivalences.

Contribution

It introduces a novel notion of projective polyhedron and offers new proofs for the equivalence of V- and H-polyhedra, enriching the mathematical foundation for applied fields.

Findings

Thorough treatment of V- and H-polytope equivalence

Introduction of the concept of projective polyhedron

New proofs for polyhedron equivalence

Abstract

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 for a broad audience of mathematically inclined readers. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also included some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of ``projective polyhedron'' based on cones. To the best of our knowledge, this notion of projective polyhedron is new. We also believe that some of our proofs establishing the equivalence of V-polyhedra and H-polyhedra are new.