Geometriae Dedicata

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

Convex equipartitions: the spicy chicken theorem

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 


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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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