Mathematika

Table of Contents - January 2012 - Volume 58, Issue 01  

Research Article

BALANCED CONVEX PARTITIONS OF MEASURES IN xs211Dd

Pablo Soberóna1

a1 Department of Mathematics, University College London, Gower Sreet, London WC1E 6BT, U.K. (email: pablo@math.unam.mx)

Abstract

We prove the following generalization of the ham sandwich theorem, conjectured by Imre Bárány. Given a positive integer k and d nice measures μ1,μ2,…,μd in xs211Dd such that μi(xs211Dd)=k for all i, there is a partition of xs211Dd into k interior-disjoint convex parts C1,C2,…,Ck such that μi (Cj)=1 for all i,j. If k=2 , this gives the ham sandwich theorem. This result was proved independently by R. N. Karasev.

(Received October 29 2010)

(Online publication October 21 2011)

MSC (2000)

  • 28A75 (primary)