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Convex equipartitions: the spicy chicken theorem. (English) Zbl 1301.52010

Summary: 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.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
28A75 Length, area, volume, other geometric measure theory
52A38 Length, area, volume and convex sets (aspects of convex geometry)
55M20 Fixed points and coincidences in algebraic topology
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