Cutting a part from many measures

Abstract

Holmsen, Kyn\\v{c}l and Valculescu recently conjectured that if a finite set $X$ with $\\ell n$ points in $\\mathbb{R}^d$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\\ell$ 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 $d$. Indeed, for integers $m, c$ and $d$ and a prime power $n=p^k$ such that $d \\geq 2$ and $m \\geq n(c-d)+\\frac{dn}{p}-\\frac{n}{p}+1$, and for $m$ positive finite absolutely continuous measures $\\mu_1, \\dots, \\mu_m$ on $\\mathbb{R}^d$, we prove that there exists a partition of $\\mathbb{R}^d$ into $n$ convex sets, such that every set has positive measure with respect to at least $c$ of the measures $\\mu_1, \\dots, \\mu_m$. Additionally, we obtain an equipartition of the measure $\\mu_m$. On the other hand, by increasing the bound on $m$ by $n-1$, we obtain an equipartition of the sum of all measures instead. The proof relies on a configuration space/test map scheme that translates this problem into a novel question from equivariant topology: We show non-existence of $\\mathfrak{S}_n$-equivariant maps from the ordered configuration space of $n$ points in $\\mathbb{R}^d$ into the union of an arrangement of affine subspaces of a Euclidean space.