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Abstract: We prove several results of the following type: any $d$ measures in $\mathbbR^d$ can be partitioned simultaneously into $k$ equal parts by a convexpartition 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. Theproofs in this paper are inspired by the generalization of the Borsuk-Ulamtheorem by M. Gromov and Y. Memarian.The main tolopogical tool in proving these facts is the lemma about thecohomology 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A newer version of this paper, merged with the similar paper of A. Hubard andB. Aronov is {arXiv:1306.2741}.



Autor: R.N. Karasev

Fuente: https://arxiv.org/







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