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.
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We are grateful to the four DCG referees for many useful comments and suggestions.
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Editor in Charge: János Pach
The authors are supported by DFG via the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.” PVMB is also supported by the Grant ON 174008 of the Serbian Ministry of Education and Science. GMZ is also supported by DFG via the Berlin Mathematical School BMS.
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Blagojević, P.V.M., Rote, G., Steinmeyer, J.K. et al. Convex Equipartitions of Colored Point Sets. Discrete Comput Geom 61, 355–363 (2019). https://doi.org/10.1007/s00454-017-9959-7
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DOI: https://doi.org/10.1007/s00454-017-9959-7