Convex equipartitions: the spicy chicken theorem

By Roman N. Karasev, Alfredo Hubard, Boris Aronov. 2013

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We show that, for any prime power $$n$$n and any convex body $$K$$K (i.e., a compact convex set with interior) in $$\mathbb{R }^d$$Rd, there exists a partition of $$K$$K into $$n$$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.

Subfields: Applied Mathematics, Computer Graphics and Computer-Aided Design, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7742c54142392e182c24b379bd473bf642c88f53