Title: Convex equipartitions of colored point sets,
Abstract: We show that any $d$-colored set of points in general position in
$\mathbb{R}^d$ can be partitioned into $n$ subsets with disjoint convex hulls
such that the set of points and all color classes are partitioned as evenly as
possible. This extends results by Holmsen, Kynčl & Valculescu (2017) and
establishes a special case of their general conjecture. Our proof utilizes a
result obtained independently by Soberón and by Karasev in 2010, on
simultaneous equipartitions of $d$ continuous measures in $\mathbb{R}^d$ by $n$
convex regions. This gives a convex partition of $\mathbb{R}^d$ with the
desired properties, except that points may lie on the boundaries of the
regions. In order to resolve the ambiguous assignment of these points, we set
up a network flow problem. The equipartition of the continuous measures gives a
fractional flow. The existence of an integer flow then yields the desired
partition of the point set.