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Balanced convex partitions of measures in \(\mathbb R^{2}\). (English) Zbl 1002.52002
Let \(n \geq 2\) be an integer and let \(\mu_1\) and \(\mu_2\) be measures in \(R^2\) such that each \(\mu_i\) is absolutely continuous with respect to the Lebesgue measure and \(\mu_1(R^2)=\mu_2(R^2)=n\).
The author shows that if \(\mu_1(B)=\mu_2(B)=n\) for some bounded domain \(B\), then there exist positive integers \(n_1\), \(n_2\) with \(n_1+n_2=n\) and disjoint open halfplanes \(D_1\), \(D_2\) such that the closure of \(D_1 \cup D_2\) is \(R^2\), \(\mu_1(D_1)= \mu_2(D_1) = n_1\) and \(\mu_1(D_2)= \mu_2(D_2) = n_2\); or there exist positive integers \(n_1\), \(n_2\), \(n_3\) with \(n_1+n_2+n_3=n\) and disjoint open convex domains \(D_1\), \(D_2\), \(D_3\) such that the closure of \(D_1 \cup D_2 \cup D_3\) is \(R^2\), \(\mu_1(D_1)= \mu_2(D_1) = n_1\), \(\mu_1(D_2)= \mu_2(D_2) = n_2\), \(\mu_1(D_3)= \mu_2(D_3) = n_3\) and such that the ray \(closure(D_1) \cap closure(D_2)\) is parallel to any non-null vector in \(R^2\).
He also shows a similar result for partitions of point sets on the plane. As a consequence of both results he obtains several corollaries giving sufficient conditions for the existence of \(k\)-convex partitions and 3-radial convex partitions.

MSC:
52A10 Convex sets in \(2\) dimensions (including convex curves)
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