Proving Tucker's Lemma with a Volume Argument

TL;DR

This paper adapts a volume-based proof technique originally used for Sperner's lemma to prove Tucker's Lemma by addressing volume distortion issues through dual polytope triangulation.

Contribution

It introduces a novel volume argument approach to prove Tucker's Lemma, extending the McLennan-Tourky technique to cross-polytopes.

Findings

Successfully proves Tucker's Lemma using volume arguments.

Addresses volume distortion by inscribing in dual polytopes.

Extends volume-based proof techniques to new topological lemmas.

Abstract

Sperner's lemma is a statement about labeled triangulations of a simplex. McLennan and Tourky (2007) provided a novel proof of Sperner's Lemma by examining volumes of simplices in a triangulation under time-linear simplex-linear deformation. We adapt a similar argument to prove Tucker's Lemma on a triangulated cross-polytope $P$. The McLennan-Tourky technique does not directly apply because this deformation may distort the volume of $P$. We remedy this by inscribing $P$ in its dual polytope, triangulating it, and considering how the volumes of deformed simplices behave.