An Introduction to Polytope Theory through Ehrhart's Theorem

Abstract

A classic introduction to polytope theory is presented, serving as the foundation to develop more advanced theoretical tools, namely the algebra of polyhedra and the use of valuations. The main theoretical objective is the construction of the so called Berline-Vergne valuation. Most of the theoretical development is aimed towards this goal. A little survey on Ehrhart positivity is presented, as well as some calculations that lead to conjecture that generalized permutohedra have positive coefficients in their Ehrhart polynomials. Throughout the thesis three different proofs of Ehrhart's theorem are presented, as an application of the new techniques developed.