A Short Proof of Euler--Poincar\'e Formula

TL;DR

This paper presents a concise, self-contained inductive proof of the Euler-Poincaré formula for convex polytopes, avoiding the use of shellability, and extends the classic Euler's formula to higher dimensions.

Contribution

It offers a new, shorter proof of the Euler-Poincaré formula that is independent of shellability, simplifying understanding of polytope topology.

Findings

Provides a concise inductive proof of the Euler-Poincaré formula

Extends the classic Euler's formula to higher-dimensional convex polytopes

Eliminates the need for shellability in the proof

Abstract

"V - E + F = 2", the famous Euler's polyhedral formula, has a natural generalization to convex polytopes in every finite dimension, also known as the Euler-Poincar\'e Formula. We provide another short inductive proof of the general formula. Our proof is self-contained and it does not use shellability of polytopes.