Convex equipartitions the spicy chicken theorem

Authors: Roman Karasev Alfredo Hubard Boris Aronov

Publish Date: 2013/07/24

Volume: 170, Issue: 1, Pages: 263-279

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Abstract

We show that for any prime power n and any convex body K ie a compact convex set with interior in mathbbR 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 hamsandwich 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 curvatureWe 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 11 Roman Karasev was supported by the Dynasty Foundation the President’s of Russian Federation grant MD35220121 the Federal Program “Scientific and scientificpedagogical staff of innovative Russia” 2009–2013 and the Russian government project 11G34310053 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 USIsrael Binational Science Foundation by NSA MSP Grant H982301010210 and by NSF Grants CCF0830691 CCF1117336 and CCF1218791

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