Convex Equipartitions: The Spicy Chicken Theorem

Understand

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 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.
• Most of the results in this paper appear in arxiv:1011.4762 and in arxiv:1010.4611.