Equipartition of several measures

TL;DR

This paper proves new equipartition results for measures and convex bodies in Euclidean spaces, using topological methods inspired by Gromov, Memarian, and Vasil'ev, partially answering longstanding geometric questions.

Contribution

It introduces novel equipartition theorems for measures and convex bodies, employing advanced topological tools and cohomology of configuration spaces.

Findings

Measures in d can be partitioned into k equal parts by convex partitions.

Convex bodies in the plane can be divided into q equal-area and perimeter parts for prime powers q.

Results partially answer open questions in geometric measure theory.

Abstract

We prove several results of the following type: any $d$ measures in $\mathbb R^d$ can be partitioned simultaneously into $k$ equal parts by a convex partition (this particular result is proved independently by Pablo Sober\'on). Another example is: Any convex body in the plane can be partitioned into $q$ parts of equal areas and perimeters provided $q$ is a prime power. The above results give a partial answer to several questions posed by A. Kaneko, M. Kano, R. Nandakumar, N. Ramana Rao, and I. B\'{a}r\'{a}ny. The proofs in this paper are inspired by the generalization of the Borsuk--Ulam theorem by M. Gromov and Y. Memarian. The main tolopogical tool in proving these facts is the lemma about the cohomology of configuration spaces originated in the work of V.A. Vasil'ev. A newer version of this paper, merged with the similar paper of A. Hubard and B. Aronov is {arXiv:1306.2741}.