2010/11/22 by Роман Карасев, Karasev, R. N. · 1 citation
Computer Science · Mathematics · #28A75 #52A38 #55R80 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1011.4762
openalex publication_date 2010/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove several results of the following type: any d measures in \mathbb Rd can be partitioned simultaneously into k equal parts by a convex partition (this particular result is proved independently by Pablo Soberón). 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. Bárány. 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 A newer version of this paper, merged with the similar paper of A. Hubard and B. Aronov is arXiv:1306.2741.