A VOLUME ARGUMENT FOR TUCKER'S LEMMA IN 2-DIMENSIONS
Beauttie Kuture1, Francis Su2.
1Pomona College, Claremont, CA, 2Harvey Mudd College, Claremont, CA.
Sperner's lemma is a combinatorial result that can be used to prove Brouwer’s fixed-point theorem and has many useful applications in economics. Recently, McLennan and Tourky provided a novel proof of Sperner's lemma using a volume argument and a linear deformation of a triangulation. We adapted a similar argument to prove Tucker's lemma on a triangulated 2-dimensional cross-polytope with the condition that its extreme points have distinct labels. However, the technique used in McLennan-Tourky’s argument does not directly apply because such deformation would distort the volume of the cross-polytope. So, we remedied this by inscribing the cross-polytope in its dual polytope, triangulating it, and considering how the volumes of the deformed simplices behave.