A survey of mass partitions

Roldán-Pensado, Edgardo; Soberón, Pablo

doi:10.1090/bull/1725

Cited by 16 publications

(7 citation statements)

References 165 publications

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“…Similar theorems have been obtained for equipartitioning continuous measures. For example, the Ham-Sandwich theorem [13] is traditionally stated in continuous setting: Given d finite absolutely continuous measures in R d , there exists a hyperplane that bisects each measure, in the sense of having half of the mass of each measure on each side; see [2] for some early history of the theorem. The 2 d -partition problem mentioned above was originally asked by Grünbaum in 1960 [7] for measures, as well; for a survey, see [3].…”

Section: Introductionmentioning

confidence: 99%

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Partitioning axis-parallel lines in 3D

Aronov¹,

Basit²,

Berg³

et al. 2022

Preprint

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Let L be a set of n axis-parallel lines in R 3 . We are are interested in partitions of R 3 by a set H of three planes such that each open cell in the arrangement A(H) is intersected by as few lines from L as possible. We study such partitions in three settings, depending on the type of splitting planes that we allow. We obtain the following results.• There are sets L of n axis-parallel lines such that, for any set H of three splitting planes, there is an open cell in A(H) that intersects at least n/3 − 1 ≈ 1 3 n lines. • If we require the splitting planes to be axis-parallel, then there are sets L of n axisparallel lines such that, for any set H of three splitting planes, there is an open cell in A(H) that intersects at least 3 2 n/4 − 1 ≈ 1 3 + 1 24 n lines. Furthermore, for any set L of n axis-parallel lines, there exists a set H of three axis-parallel splitting planes such that each open cell in A(H) intersects at most 7 18 n = 1 3 + 1 18 n lines. • For any set L of n axis-parallel lines, there exists a set H of three axis-parallel and mutually orthogonal splitting planes, such that each open cell in A(H) intersects at most 5 12 n ≈ 1 3 + 1 12 n lines.

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Section: Introductionmentioning

confidence: 99%

“…The 2 d -partition problem mentioned above was originally asked by Grünbaum in 1960 [7] for measures, as well; for a survey, see [3]. For an overview of equipartitioning problems, see [13,15].…”

Section: Introductionmentioning

confidence: 99%

Partitioning axis-parallel lines in 3D

Aronov¹,

Basit²,

Berg³

et al. 2022

Preprint

View full text Add to dashboard Cite

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“…Determining whether such partitions always exist leads to a rich family of problems. Solutions to these problems often require topological methods and can have computational applications [Mat03,Živ17,RPS21]. The quintessential mass partition result is the ham sandwich theorem, conjectured by Steinhaus and proved by Banach [Ste38].…”

Section: Introductionmentioning

confidence: 99%

Lifting methods in mass partition problems

Soberón¹,

Takahashi²

2021

Preprint

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Many results in mass partitions are proved by lifting R d to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other results, we prove the existence of equipartitions of d + 1 measures in R d by parallel hyperplanes and of d + 2 measures in R d by concentric spheres.For measures whose supports are sufficiently well separated, we prove results where one can cut a fixed (possibly different) fraction of each measure either by parallel hyperplanes, concentric spheres, convex polyhedral surfaces of few facets, or convex polytopes with few vertices.

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“…Mass partition problems study how one can split finite sets of points or measures in Euclidean spaces. They connect topological combinatorics and computational geometry [Mat03,Živ17,RPS21]. We say that a finite family P of subsets of R d is a convex partition of R d if the union of the sets is R d , the interiors of the sets are pairwise disjoint, and each set is closed and convex.…”

Section: Introductionmentioning

confidence: 99%

Generalizations of the Yao-Yao partition theorem and the central transversal theorem

Manta,

Soberón

2021

Preprint

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We generalize the Yao-Yao partition theorem by showing that for any smooth measure in R d there exist equipartitions using (t + 1)2 d−1 convex regions such that every hyperplane misses the interior of at least t regions. In addition, we present tight bounds on the smallest number of hyperplanes whose union contains the boundary of an equipartition of a measure into n regions. We also present a simple proof of a Borsuk-Ulam type theorem for Stiefel manifolds that allows us to generalize the central transversal theorem and prove results bridging the Yao-Yao partition theorem and the central transversal theorem.

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A survey of mass partitions

Cited by 16 publications

References 165 publications

Partitioning axis-parallel lines in 3D

Partitioning axis-parallel lines in 3D

Lifting methods in mass partition problems

Generalizations of the Yao-Yao partition theorem and the central transversal theorem

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