How the g-Conjecture Came About

This post complements Eran Nevo’s first  post on the g-conjecture

1) Euler’s theorem

euler1

Euler

Euler’s famous formula V-E+F=2 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 d-dimensional polytopes P is

f_0(P)-f_1(P)+f_2(P)+\dots+(-1)^{d-1}f_{d-1}(P)=1-(-1)^d.

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

Dehn

Consider a 3-dimensional simplicial polytope, – Continue reading