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arXiv · 2107.06233

Generalizations of the Yao-Yao partition theorem and the central transversal theorem

Michael N. Manta·Pablo Soberón
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Abstract

We generalize the Yao-Yao partition theorem by showing that for any smooth measure in $R^d$ there exist equipartitions using $(t+1)2^{d-1}$ convex regions such that every hyperplane misses the interior of at least $t$ regions. In addition, we present tight bounds on the smallest number of hyperplanes whose union contains the boundary of an equipartition of a measure into $n$ regions. We also present a simple proof of a Borsuk-Ulam type theorem for Stiefel manifolds that allows us to generalize the central transversal theorem and prove results bridging the Yao--Yao partition theorem and the central transversal theorem.

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Michael N. Manta, Pablo Soberón. 2021-07-13. Generalizations of the Yao-Yao partition theorem and the central transversal theorem. https://arxiv.org/abs/2107.06233

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