Proving Tucker's Lemma with a Volume Argument
Combinatorics
2016-04-11 v1 Algebraic Topology
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 . The McLennan-Tourky technique does not directly apply because this deformation may distort the volume of . We remedy this by inscribing in its dual polytope, triangulating it, and considering how the volumes of deformed simplices behave.
Cite
@article{arxiv.1604.02395,
title = {Proving Tucker's Lemma with a Volume Argument},
author = {Beauttie Kuture and Oscar Leong and Christopher Loa and Mutiara Sondjaja and Francis Edward Su},
journal= {arXiv preprint arXiv:1604.02395},
year = {2016}
}