Abstract
We generalize the Yao–Yao partition theorem by showing that for any smooth measure in \(R^d\) there exist equipartitions using \((t+1)\hspace{0.49988pt}2^{d-1}\) 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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The authors thank the two anonymous referees whose comments significantly improved this manuscript.
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Soberón’s research is supported by NSF Grant DMS 2054419 and a PSC-CUNY TRADB52 award.
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Manta, M.N., Soberón, P. Generalizations of the Yao–Yao Partition Theorem and Central Transversal Theorems. Discrete Comput Geom 71, 1381–1402 (2024). https://doi.org/10.1007/s00454-023-00536-7
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DOI: https://doi.org/10.1007/s00454-023-00536-7