Convex equipartitions: The spicy chicken theorem

Roman Karasev, Alfredo Hubard, Boris Aronov

    Research output: Contribution to journalArticle

    Abstract

    We show that, for any prime power n and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of 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.

    Original languageEnglish (US)
    Pages (from-to)263-279
    Number of pages17
    JournalGeometriae Dedicata
    Volume170
    Issue number1
    DOIs
    StatePublished - 2014

    Fingerprint

    Equipartition
    Convex Body
    Convex Sets
    Borsuk-Ulam Theorem
    Theorem
    Partition
    Compact Convex Set
    Sandwich
    Absolutely Continuous
    Surface area
    Interior
    Curvature
    Arbitrary
    Generalization
    Model

    Keywords

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

    ASJC Scopus subject areas

    • Geometry and Topology

    Cite this

    Convex equipartitions : The spicy chicken theorem. / Karasev, Roman; Hubard, Alfredo; Aronov, Boris.

    In: Geometriae Dedicata, Vol. 170, No. 1, 2014, p. 263-279.

    Research output: Contribution to journalArticle

    Karasev, R, Hubard, A & Aronov, B 2014, 'Convex equipartitions: The spicy chicken theorem', Geometriae Dedicata, vol. 170, no. 1, pp. 263-279. https://doi.org/10.1007/s10711-013-9879-5
    Karasev, Roman ; Hubard, Alfredo ; Aronov, Boris. / Convex equipartitions : The spicy chicken theorem. In: Geometriae Dedicata. 2014 ; Vol. 170, No. 1. pp. 263-279.
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