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 (Comput Geom 65:35–42, 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.