  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.
