3 research outputs found
KKM-type theorems for products of simplices and cutting sets and measures by straight lines
In this paper a version of Knaster-Kuratowski-Mazurkiewicz theorem for
products of simplices is formulated. Some corollaries for measure partition in
the plane and cutting families of sets in the plane by lines are given.Comment: This paper was not accepted for publication in a journal. The referee
considered the main topological lemma known. Though some corollaries may be
of value, so this paper is posted to arxiv.or
Equipartition of several measures
We prove several results of the following type: any measures in can be partitioned simultaneously into 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
parts of equal areas and perimeters provided 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A newer version of this paper, merged with the similar paper of A. Hubard and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}
Convex Equipartitions: The Spicy Chicken Theorem
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.Comment: Appendix is added answering the criticism that appeared after the
paper was published in Geometriae Dedicat