Abstract.
Let n≥2 be an integer and let μ1 and μ2 be measures in ℝ2 such that each μ i is absolutely continuous with respect to the Lebesgue measure and μ1(ℝ2)=μ2(ℝ2)=n. Let u≠0 be a vector on the plane. We show that if μ1(B)=μ2(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 half-planes D 1,D 2 such that , μ1(D 1)=μ2(D 1)=n 1 and μ1(D 2)=μ2(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 , μ1(D 1)=μ2(D 1)=n 1, μ1(D 2)= μ2(D 2)=n 2, μ1(D 3)=μ2(D 3)=n 3 and such that the ray is parallel to u. We also show a similar result for partitioning of point sets on the plane.
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Received: November 24, 1999 Final version received: February 9, 2001
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Sakai, T. Balanced Convex Partitions of Measures in ℝ2. Graphs Comb 18, 169–192 (2002). https://doi.org/10.1007/s003730200011
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DOI: https://doi.org/10.1007/s003730200011