Proving Tucker's Lemma with a volume argument

摘要

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.