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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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