Geometriae Dedicata

, Volume 170, Issue 1, pp 263–279 | Cite as

Convex equipartitions: the spicy chicken theorem

  • Roman Karasev
  • Alfredo Hubard
  • Boris Aronov
Original Paper


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.


Equipartitions Waist Borsuk–Ulam  Ham sandwich Voronoi diagram Nandakumar–Ramana Rao conjecture Configuration space 

Mathematics Subject Classification

28A75 52A38 55M20 



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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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.Institute for Information Transmission Problems RASMoscowRussia
  3. 3.Laboratory of Discrete and Computational GeometryYaroslavl State UniversityYaroslavlRussia
  4. 4.Département d’InformatiqueÉcole Normale SupérieureParisFrance
  5. 5.Department of Computer Science and EngineeringPolytechnic Institute of NYUBrooklynUSA

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