This post complements Eran Nevo’s first post on the -conjecture
1) Euler’s theorem
Euler’s famous formula for the numbers of vertices, edges and faces of a polytope in space is the starting point of many mathematical stories. (Descartes came close to this formula a century earlier.) The formula for -dimensional polytopes is
The first complete proof (in high dimensions) was provided by Poincare using algebraic topology. Earlier geometric proofs were based on “shellability” of polytopes which was only proved a century later. But there are elementary geometric proofs that avoid shellability.
2) The Dehn-Sommerville relations
Consider a 3-dimensional simplicial polytope, – Continue reading