Equipartitions of Measures by 2-Fans

Bereg, Sergey

doi:10.1007/s00454-004-1151-1

Cited by 19 publications

(10 citation statements)

References 21 publications

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“…These results appear in Sect. 4. Also, as a corollary to Theorem 1, we observe that in the discrete case, the conditions for the existence and uniqueness of all cuts are also necessary.…”

Section: Definition 3 Points In Smentioning

confidence: 83%

“…For example, Lo et al [9] gave a direct proof of a discrete version of the ham-sandwich theorem which inspired an efficient algorithm to compute a cut. More recently Bereg [4] studied a discrete version of a result of Bárány and Matoušek [2] that showed the existence of wedges that simultaneously equipartition three measures on R 2 (they are called equitable two-fans). By seeking a direct, combinatorial proof of a discrete version (for counting measure on points sets in R 2 ) he was able to strengthen the original result and also obtained a beautiful, nearly optimal algorithm to construct an equitable two-fan.…”

mentioning

confidence: 99%

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Generalized Ham-Sandwich Cuts

Steiger

Zhao

2009

Discrete Comput Geom

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Bárány, Hubard, and Jerónimo recently showed that for given wellseparated convex bodies S 1 , . . . , S d in R d and constants β i ∈ [0, 1], there exists a unique hyperplane h with the property that Vol(h + ∩ S i ) = β i · Vol(S i ); h + is the closed positive transversal halfspace of h, and h is a "generalized ham-sandwich cut." We give a discrete analogue for a set S of n points in R d which are partitioned into a family S = P 1 ∪ · · · ∪ P d of well-separated sets and are in weak general position. The combinatorial proof inspires an O(n(log n) d−3 ) algorithm which, given positive integers a i ≤ |P i |, finds the unique hyperplane h incident with a point in each P i and having |h + ∩ P i | = a i . Finally we show two other consequences of the direct combinatorial proof: the first is a stronger result, namely that in the discrete case, the conditions assuring existence and uniqueness of generalized cuts are also necessary; the second is an alternative and simpler proof of the theorem in Bárány et al., and in addition, we strengthen the result via a partial converse.

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“…These results appear in Sect. 4. Also, as a corollary to Theorem 1, we observe that in the discrete case, the conditions for the existence and uniqueness of all cuts are also necessary.…”

Section: Definition 3 Points In Smentioning

confidence: 83%

mentioning

confidence: 99%

Generalized Ham-Sandwich Cuts

Steiger

Zhao

2009

Discrete Comput Geom

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“…This curse of dimensionality seems to be a common issue for many algorithmic version of mass partition problems, and most problems have only been studied from an algorithmic point of view in low dimensions, where the constructed algorithms either rely on a relatively small space of solutions or a simplified proof which allows for an algorithmic formulation, see e.g. [1,6,37].…”

Section: Algorithms and Complexitymentioning

confidence: 99%

The complexity of sharing a pizza

Schnider

2021

Preprint

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Assume you have a 2-dimensional pizza with 2n ingredients that you want to share with your friend. For this you are allowed to cut the pizza using several straight cuts, and then give every second piece to your friend. You want to do this fairly, that is, your friend and you should each get exactly half of each ingredient. How many cuts do you need?It was recently shown using topological methods that n cuts always suffice. In this work, we study the computational complexity of finding such n cuts. Our main result is that this problem is PPA-complete when the ingredients are represented as point sets. For this, we give a new proof that for point sets n cuts suffice, which does not use any topological methods.We further prove several hardness results as well as a higher-dimensional variant for the case where the ingredients are well-separated.

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“…The fact that d + 1 measures in R d can be halved by a wedge was first proved by Bárány and Matoušek in dimension two [BM01] and later generalized to R d by Schnider [Sch19]. For R 2 , Bereg presented algorithmic approaches for the discrete version which show that more conditions can be imposed to the wedge [Ber05].…”

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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Equipartitions of Measures by 2-Fans

Cited by 19 publications

References 21 publications

Generalized Ham-Sandwich Cuts

Generalized Ham-Sandwich Cuts

The complexity of sharing a pizza

Lifting methods in mass partition problems

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