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Generalizations of the Yao–Yao Partition Theorem and Central Transversal Theorems

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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 \((t+1)\hspace{0.49988pt}2^{d-1}\) 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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Acknowledgements

The authors thank the two anonymous referees whose comments significantly improved this manuscript.

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Authors and Affiliations

  1. California Institute of Technology, Pasadena, CA, 91125, USA

    Michael N. Manta

  2. Baruch College & The Graduate Center, City University of New York, New York, NY, 10010, USA

    Pablo Soberón

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  1. Michael N. Manta
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  2. Pablo Soberón
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Correspondence to Pablo Soberón.

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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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