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

    Keywords

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

    ASJC Scopus subject areas

    • Geometry and Topology

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