Cutting a part from many measures

Abstract

Holmsen, Kynčl and Valculescu recently conjectured that if a finite set X with ℓn points in Rd that is colored by m different colors can be partitioned into n subsets of ℓ points each, such that each subset contains points of at least d different colors, then there exists such a partition of X with the additional property that the convex hulls of the n subsets are pairwise disjoint. We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least c different colors, where we also allow c to be greater than d. Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from c different colors. For example, when n⩾2, d⩾2, c⩾d with m⩾n(c−d)+d are integers, and μ1,…,μm are m positive finite absolutely continuous measures on Rd, we prove that there exists a partition of Rd into n convex pieces which equiparts the measures μ1,…,μd−1, and in addition every piece of the partition has positive measure with respect to at least c of the measures μ1,…,μm.