Convex Equipartitions: The Spicy Chicken Theorem

TL;DR

This paper proves that any convex body can be partitioned into an equal number of convex parts with identical volume and surface area, extending classical theorems and confirming longstanding conjectures in convex geometry.

Contribution

It introduces new equipartition results for convex bodies with equal volume and surface area, generalizing the ham-sandwich theorem and Gromov-Borsuk-Ulam theorem for convex sets.

Findings

Partition of convex bodies into n equal-volume, equal-surface-area convex sets.

Generalization of the ham-sandwich theorem to multiple convex pieces.

Extension of the Gromov-Borsuk-Ulam theorem for convex sets.

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 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. Since the main results and techniques there are essentially the same, we have merged the papers for journal publication. In this version we also provide a technical alternative to a part of the proof of the main topological result that avoids the use of compactly supported homology.