Equipartition of several measures

Details Equipartition of several measures Equipartition of several measures

2010-11-22

arxiv.org/abs/1011.4762v6

Mathematics

Metric Geometry

Scientific paper

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 P. 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\'ar\'any. The proofs in this paper are inspired by the generalization of the Borsuk-Ulam theorem by M. Gromov and Y. Memarian.

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