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Convex equipartitions of colored point sets. (English) Zbl 1407.52031

Summary: 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 A. F. Holmsen et al. [Comput. Geom. 65, 35–42 (2017; Zbl 1377.65026)] and establishes a special case of their general conjecture. Our proof utilizes a result obtained independently by P. Soberón [Mathematika 58, No. 1, 71–76 (2012; Zbl 1267.28005)] and by R. N. Karasev in [“Equipartition of several measures”, Preprint, arXiv:1011.4762], 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.

MSC:

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
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