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 |
Keywords:
equipartitions; waist; Borsuk-Ulam; ham sandwich; Voronoi diagram; Nandakumar-Ramana Rao conjecture; configuration spaceReferences:
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