Abstract
We show that, for any prime power \(n\) and any convex body \(K\) (i.e., a compact convex set with interior) in \(\mathbb{R }^d\), there exists a partition of \(K\) into \(n\) convex sets with equal volumes and equal surface areas. Similar results regarding equipartitions with respect to continuous functionals and absolutely continuous measures on convex bodies are also proven. These include a generalization of the ham-sandwich theorem to arbitrary number of convex pieces confirming a conjecture of Kaneko and Kano, a similar generalization of perfect partitions of a cake and its icing, and a generalization of the Gromov–Borsuk–Ulam theorem for convex sets in the model spaces of constant curvature.
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Notes
Vegetarian readers are welcome to substitute the chicken filet with a peeled potato.
The term “power” comes from Euclidean geometry. Recall that the power of a point \(p\) with respect to a circle of radius \(r\) and center \(y\), that does not contain \(p\), is \(|p-y|^2-r^2\).
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Acknowledgments
We thank Arseniy Akopyan, Imre Bárány, Pavle Blagojević, Sylvain Cappell, Fred Cohen, Daniel Klain, Erwin Lutwak, Yashar Memarian, Ed Miller, Gabriel Nivasch, Steven Simon, and Alexey Volovikov for discussions, useful remarks, and references. We also thank an anonymous referee for encouraging us to merge our papers and for his/her enthusiasm towards the chicken nuggets description of Corollary 1.1. Roman Karasev was supported by the Dynasty Foundation, the President’s of Russian Federation grant MD-352.2012.1, the Federal Program “Scientific and scientific-pedagogical staff of innovative Russia” 2009–2013, and the Russian government project 11.G34.31.0053. Boris Aronov and Alfredo Hubard gratefully acknowledge the support of the Centre Interfacultaire Bernoulli at EPFL, Lausanne, Switzerland. Alfredo Hubard thankfully acknowledges the support from CONACyT and from the Fondation Sciences Matheḿatiques de Paris. The research of Boris Aronov has been supported in part by a grant No. 2006/194 from the U.S.-Israel Binational Science Foundation, by NSA MSP Grant H98230-10-1-0210, and by NSF Grants CCF-08-30691, CCF-11-17336, and CCF-12-18791.
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Karasev, R., Hubard, A. & Aronov, B. Convex equipartitions: the spicy chicken theorem. Geom Dedicata 170, 263–279 (2014). https://doi.org/10.1007/s10711-013-9879-5
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DOI: https://doi.org/10.1007/s10711-013-9879-5
Keywords
- Equipartitions
- Waist
- Borsuk–Ulam
- Ham sandwich
- Voronoi diagram
- Nandakumar–Ramana Rao conjecture
- Configuration space