A Volume Argument for Tucker's Lemma

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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