Balanced Convex Partitions of Measures in ℝ2
Balanced Convex Partitions of Measures in ℝ2
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ℝ2 中测度的平衡凸划分
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
T. Sakai
中科院分区：
文献类型：
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作者：
T. Sakai
设n≥2是整数，μ1和μ2是μ 2中的测度，使得μ 1关于Lebesgue测度绝对连续，μ1（μ 2）=μ2（μ 2）=n.设u ∈ 0是平面上的一个向量。本文证明了：对于某个有界区域B，若μ1（B）=μ2（B）=n，则存在正整数n1，n2，且n1+n2=n，存在不相交的开半平面D1，D2，使得μ1（D1）=μ2（D1）=n1，μ1（D2）=μ2（D2）=n2;或者存在正整数n1，n2，n3，其中n1+n2+n3=n，且存在不相交的开凸域D1，D2，D3，使得μ1（D1）=μ2（D1）=n1，μ1（D2）= μ2（D2）=n2，μ1（D3）=μ2（D3）=n3，且光线平行于u。我们也显示了类似的结果划分点集的平面上。
Abstract. Let n≥2 be an integer and let μ1 and μ2 be measures in ℝ2 such that each μi 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 n1,n2 with n1+n2=n and disjoint open half-planes D1,D2 such that , μ1(D1)=μ2(D1)=n1 and μ1(D2)=μ2(D2)=n2; or there exist positive integers n1,n2,n3 with n1+n2+n3=n and disjoint open convex domains D1,D2,D3 such that , μ1(D1)=μ2(D1)=n1, μ1(D2)= μ2(D2)=n2, μ1(D3)=μ2(D3)=n3 and such that the ray is parallel to u. We also show a similar result for partitioning of point sets on the plane.