Sperner's lemma is a statement about labeled triangulations of a simplex.McLennan and Tourky (2007) provided a novel proof of Sperner's Lemma byexamining volumes of simplices in a triangulation under time-linearsimplex-linear deformation. We adapt a similar argument to prove Tucker's Lemmaon a triangulated cross-polytope $P$. The McLennan-Tourky technique does notdirectly apply because this deformation may distort the volume of $P$. Weremedy this by inscribing $P$ in its dual polytope, triangulating it, andconsidering how the volumes of deformed simplices behave.
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机译：Sperner的引理是关于标记的单纯形三角剖分的陈述.McLennan和Tourky（2007）通过检查时间-线性-简单-线性变形下三角剖分中的单纯形体的体积，提供了Sperner引理的新颖证明。我们采用了类似的论点，以证明塔克的引理是一个三角形的交叉多面体$ P $。 McLennan-Tourky技术不能直接应用，因为这种变形可能会使$ P $的体积变形。我们通过将$ P $刻在其双重多面体中，对其进行三角剖分并考虑变形后的单形体的行为方式来对此进行补救。
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