2021
DOI: 10.1090/bull/1725
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A survey of mass partitions

Abstract: Mass partition problems describe the partitions we can induce on a family of measures or finite sets of points in Euclidean spaces by dividing the ambient space into pieces. In this survey we describe recent progress in the area in addition to its connections to topology, discrete geometry, and computer science.

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Cited by 17 publications
(7 citation statements)
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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%
See 1 more Smart Citation
“…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%
“…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%
“…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%
“…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%