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A Volume Argument for Tucker's Lemma

MSRI-UP 2015: Geometric Combinatorics Motivated by the Social Sciences June 13, 2015 - July 26, 2015

July 24, 2015 (02:45 PM PDT - 03:30 PM PDT)
Speaker(s): Beauttie Kuture (Pomona College), Oscar Leong (Swarthmore College), Christopher Loa (University of Tennessee)
Location: MSRI: Baker Board Room
Video

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 using a volume argument and a piecewise linear deformation of a triangulation. We adapt a similar argument to prove Tucker’s Lemma on a triangulated cross-polytope P in the 2-dimensional case where vertices of P have different labels. TheMcLennan-Tourky technique would not directly apply because the natural deformation distorts 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.

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Kuture, Leong, Loa

H.264 Video Team_6.mp4 116 MB video/mp4 rtsp://videos.msri.org/data/000/023/920/original/Team_6.mp4 Download
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