Cutting a part from many measures

TL;DR

This paper proves a continuous analogue of a conjecture on partitioning colored point sets in Euclidean space, ensuring convex partitions with measure and color diversity, and provides bounds on the measure fractions in each subset.

Contribution

It generalizes a conjecture by Holmsen, Kynčl, and Valculescu to continuous measures, allowing for higher color diversity and measure bounds in convex partitions.

Findings

Established existence of convex partitions with measure and color diversity.

Provided lower bounds on the measure fractions for each color in the partitions.

Extended the conjecture to continuous measures with generalized parameters.

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 $d$. Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geq 2$, $d\geq 2$, $c\geq d$ with $m\geq n(c-d)+d$ are integers, and $\mu_1, \dots, \mu_m$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^d$, we prove that there exists a partition of $\mathbb{R}^d$ into $n$ convex pieces which equiparts the measures $\mu_1, \dots, \mu_{d-1}$, and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $\mu_1, \dots, \mu_m$.