a1 Department of Mathematics, University College London, Gower Sreet, London WC1E 6BT, U.K. (email: email@example.com)
We prove the following generalization of the ham sandwich theorem, conjectured by Imre Bárány. Given a positive integer k and d nice measures μ 1,μ 2,…,μ d in ℝ d such that μ i (ℝ d )=k for all i, there is a partition of ℝ d into k interior-disjoint convex parts C 1,C 2,…,C k such that μ i (C j )=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)