Multilabeled Versions of Sperner's and Fan's Lemmas and Applications

Meunier, Frédéric; Su, Francis Edward

doi:10.1137/18m1192548

Cited by 13 publications

(12 citation statements)

References 31 publications

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“…Hence, the possibility of constructing ad hoc groups plays a crucial role in ensuring that each group receives a contiguous piece and all players are nonenvious. 2 Theorem 2 can be shown by applying a generalized Sperner-type lemma similar to the one proved recently by Meunier and Su [10]. However, here we present a simpler proof using Theorem 1.…”

Section: Our Resultmentioning

confidence: 71%

How to Cut a Cake Fairly: A Generalization to Groups

Segal-Halevi,

Suksompong

2020

Preprint

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A fundamental result in cake cutting states that for any number of players with arbitrary preferences over a cake, there exists a division of the cake such that every player receives a single contiguous piece and no player is left envious. We generalize this result by showing that it is possible to partition the players into groups of any desired sizes and divide the cake among the groups so that each group receives a single contiguous piece, and no player finds the piece of another group better than that of the player's own group.1. Hungry players. Players never prefer an empty piece. 2. Closed preference sets. Any piece that is preferred for a convergent sequence of partitions is also preferred at the limiting partition.

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Section: Our Resultmentioning

confidence: 71%

How to Cut a Cake Fairly: A Generalization to Groups

Segal-Halevi,

Suksompong

2020

Preprint

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“…Meunier and Su proved other results for cake splitting with more guests than cake pieces [MS19]. One of their results shows that the cake can be divided into k pieces in advance of the n guests arriving, and even if (n − k)/k guests miss the party, we can distribute the cake in an envy-free way among k of those who showed up.…”

Section: Introductionmentioning

confidence: 83%

Fair distributions for more participants than allocations

Soberón¹

2021

Preprint

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We study the existence of fair distributions when we have more players than pieces to allocate, focusing on envy-free distributions among those who receive a piece. The conditions on the demand from the players can be weakened from those of classic cake-cutting and rent-splitting results of Stromquist, Woodall, and Su. We extend existing variations of the cakesplitting problem with secretive players and those that resist the removal of any sufficiently small set of players.

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“…As a corollary, we obtain the already known multilabeled version of Sperner's lemma. For a purely combinatorial proof and a more elaborate discussion, one can see [6], however one should note the subtle differences between statements given in this chapter and theorems in [6] -we are dealing with triangulated balls instead of arbitrary free Z 2complexes and our results claim odd number of simplices of a given type (instead of merely their existence).…”

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confidence: 88%

Homotopies and transcendental extensions in colouring problems

Duliński

2020

Preprint

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The goal of this paper is to show some new applications and possible generalizations of the oriented volume method applied to the polytope colouring problems. This approach originated from the work of McLennan and Tourky ([1]), several sources attribute some role in the development of this technique to John Milnor as well. The advantage of this method lies in its geometric nature, appealing to the intuitions from the theory of manifolds and from elementary notions of the homotopy theory. We use some well-known algebraic formulas for quantities related to the volume of simplices (generalized shoelace formula, Cayley-Menger determinant).We prove new results using this method: the non-draw property of the generalized Y game, the theorem about triangulation of the product of two simplices, the generalized Atanassov conjecture, and give new proofs of known results: the multilabeled versions of Sperner's and Ky Fan's lemmas.We also develop the technique of geometric realizations with algebraically independent (over the filed of real algebraic numbers) coordinates of vertices. This enables us to link the method of oriented volume and the proof of Ky Fan's lemma, and also between statements on linear combinations of multivariate polynomials and theorems about colourings of polytopes.We conclude with some remarks about possible further developments. Motivation and basic notions.In this paper we are concerened primarily with triangulated polytopes and colourings of their vertices. By a n-simplex ∆ n we understand a convex hull of n + 1 affinely independent points p 1 , . . . , p n+1 in a Euclidean space. By its d-face we mean a convex hull of d + 1 distinct points from {p 1 , . . . , p n+1 }. If we don't want to specify the exact number of these points, we simply speak of a face of a simplex.

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Multilabeled Versions of Sperner's and Fan's Lemmas and Applications

Cited by 13 publications

References 31 publications

How to Cut a Cake Fairly: A Generalization to Groups

How to Cut a Cake Fairly: A Generalization to Groups

Fair distributions for more participants than allocations

Homotopies and transcendental extensions in colouring problems

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