Sharing a pizza: bisecting masses with two cuts

Barba, Luis; Pilz, Alexander; Schnider, Patrick

doi:10.48550/arxiv.1904.02502

Cited by 6 publications

(7 citation statements)

References 19 publications

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“…In the first part of this paper we consider arrangements of lines in the projective plane and investigate the average complexity of the k-level, when the arrangement is "randomly" projected to an Euclidean arrangement. This question was raised by Barba, Pilz, and Schnider while sharing a pizza [BPS19].…”

Section: Our Resultsmentioning

confidence: 99%

On the Average Complexity of the $k$-Level

Chiu¹,

Felsner²,

Scheucher³

et al. 2019

Preprint

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Let A be an arrangement of n lines in the Euclidean plane. The k-level of A consists of all intersection points v of lines in A which have exactly k lines of A passing below v. The complexity of the k-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of O(n • (k + 1) 1/3 ). We investigate the complexity of k-levels in random line and hyperplane arrangements. When the arrangement is obtained from any fixed projective line arrangement of n lines by choosing a random cell to contain the south-pole, we prove an upper bound of O((k + 1) 2 ) on the expected complexity of the k-level. As a byproduct we show that the complexity of any (≤ j)-zone in a d-dimensional simple arrangement of n hyperplanes is of order Θ((j + 1)n d−1 ). The classical zone theorem is the case j = 0.We also consider arrangements of great (d − 1)-spheres on the sphere S d which are orthogonal to a set of random points on S d . In this model we prove that the expected complexity of the k-level is of order Θ((k + 1) d−1 ).

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Section: Our Resultsmentioning

confidence: 99%

On the Average Complexity of the $k$-Level

Chiu¹,

Felsner²,

Scheucher³

et al. 2019

Preprint

Self Cite

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“…For the other changes, we need the following lemma. Here, we say that an arrangement A (k. 5) is almost bisecting for P (k.5) , if it bisects each P (k.5) i except for one, for which two points are on a line of the arrangement, and for the remaining points one side contains exactly one point more.…”

Section: A Proof Of the Discrete Planar Pizza Cutting Theoremmentioning

confidence: 99%

“…It was conjectured by Langerman that any nd mass distributions in R d can be simultaneously bisected by an arrangement of n hyperplanes ( [30], see also [5]). In a series of papers, this conjecture has been resolved for 4 masses in the plane [5], for any number of masses in any dimension that is a power of 2 (and thus in particular also in the plane) [26] and in a relaxed setting for any number of masses in any dimension [41]. However, the general conjecture remains open.…”

Section: Introduction 1mass Partitionsmentioning

confidence: 99%

The complexity of sharing a pizza

Schnider

2021

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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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“…Theorem 1.2 is also related to the problem of halving measures in R d using hyperplane arrangements. Langerman conjectured that any dn measures in R d can be simultaneously halved by a chessboard coloring induced by n hyperplanes [BPS19,HK20]. For n = 2, this has been confirmed for 2d − O(log d) measures [BBKK18].…”

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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Sharing a pizza: bisecting masses with two cuts

Cited by 6 publications

References 19 publications

On the Average Complexity of the $k$-Level

On the Average Complexity of the $k$-Level

The complexity of sharing a pizza

Lifting methods in mass partition problems

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