Fair Division and Generalizations of Sperner- and KKM-type Results

Asada, Megumi; Frick, Florian; Pisharody, Vivek; Polevy, Maxwell; Stoner, David; Tsang, Ling Hei; Wellner, Zoe

doi:10.1137/17m1116210

Cited by 23 publications

(36 citation statements)

References 32 publications

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“…In light of the recent counterexamples to the topological Tverberg conjecture for parameters that are not prime powers [5,11,18], this opens the interesting question of whether the primality of r is perhaps not an artifact of our proof method, but actually an essential prerequisite of our result. Asada et al [4], and Blagojević and Soberón [6]. Here we show the following: Let C 1 , .…”

Section: Necklace Splittings With Additional Constraintssupporting

confidence: 57%

“…That is, instead of the sides of the rectangle itself having the same length, we can only ensure this for the pieces of the loop over those sides. rectangle cutting it into four pieces γ (1) , γ (2) , γ (3) , γ (4) in cyclic order such that γ (1) and γ (3) have the same total length as γ (2) and γ (4) .…”

Section: Splitting Rectifiable Loopsmentioning

confidence: 99%

“…, γ (4) in cyclic order such that the total length of γ (1) and γ (3) is equal to the total length of γ (2) and γ (4) ; see Theorem 3.2.…”

Section: Introductionmentioning

confidence: 99%

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Splitting Loops and Necklaces: Variants of the Square Peg Problem

Aslam

Chen

Frick

et al. 2020

Forum of Mathematics, Sigma

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Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz' square peg problem. We prove Hadwiger's 1971 conjecture that any simple loop in 3-space inscribes a parallelogram. We show that any simple planar loop inscribes sufficiently many rectangles that their vertices are dense in the loop (independently due to Schwartz). If the loop is rectifiable, there is a rectangle that cuts the loop into four pieces that can be rearranged to form two loops of equal length. A rectifiable loop in d-space can be cut into (r − 1)(d + 1) + 1 pieces that can be rearranged by translations to form r loops of equal length.We relate our results to fair divisions of necklaces in the sense of Alon and to Tverberg-type results. This provides a new approach and a common framework to obtain variants of Toeplitz' square peg problem for the class of all continuous curves.

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Section: Necklace Splittings With Additional Constraintssupporting

confidence: 57%

Section: Splitting Rectifiable Loopsmentioning

confidence: 99%

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Splitting Loops and Necklaces: Variants of the Square Peg Problem

Aslam

Chen

Frick

et al. 2020

Forum of Mathematics, Sigma

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“…We also prove -with a completely different approach -the following generalization of this lemma, which can be seen as a homological counterpart of the polytopal Sperner lemma by De Loera, Peterson, and Su [5], in a same way Meshulam's lemma is a homological countepart of the classical Sperner lemma. It can also be seen as a homological counterpart of Musin's Sperner-type results for pseudomanifolds [15] and of Theorem 4.7 in the paper by Asada et al [2]. We leave as an open question the existence of a proof based on a nerve theorem of some kind.We recall that a pseudomanifold is a simplicial complex that is pure, non-branching (each ridge is contained in exactly two facets), and strongly connected (the dual is connected).…”

mentioning

confidence: 85%

Different versions of the nerve theorem and colourful simplices

Meunier

Montejano²

2020

Journal of Combinatorial Theory, Series A

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Given a simplicial complex and a collection of subcomplexes covering it, the nerve theorem, a fundamental tool in topological combinatorics, guarantees a certain connectivity of the simplicial complex when connectivity conditions on the intersection of the subcomplexes are satisfied.We show that it is possible to extend this theorem by replacing some of these connectivity conditions on the intersection of the subcomplexes by connectivity conditions on their union. While this is interesting for its own sake, we use this extension to generalize in various ways the Meshulam lemma, a powerful homological version of the Sperner lemma. We also prove a generalization of the Meshulam lemma that is somehow reminiscent of the polytopal generalization of the Sperner lemma by De Loera, Peterson, and Su. For this latter result, we use a different approach and we do not know whether there is a way to get it via a nerve theorem of some kind. versions, some of them with a connectivity condition in place of the acyclicity condition. It seems that the oldest reference to a nerve theorem with an acyclicity condition is due to Leray [9]. The case k = −1 is the nerve theorem for unions mentioned above.The second purpose of this paper is to provide a generalization of a related result -Meshulam's lemma [11, Proposition 1.6] and [10, Theorem 1.5] -which has several applications in combinatorics, such as the generalization of Edmonds' intersection theorem by Aharoni and Berger [1]. Meshulam's lemma is a Sperner-lemma type result, dealing with coloured simplicial complexes and colourful simplices, and in which the classical boundary condition of the Sperner lemma is replaced by an acyclicity condition. Its original proof relies on a certain version of the nerve theorem and we show that Theorem 1 can be used in the same vein to prove some variations of Meshulam's lemma. We also prove -with a completely different approach -the following generalization of this lemma, which can be seen as a homological counterpart of the polytopal Sperner lemma by De Loera, Peterson, and Su [5], in a same way Meshulam's lemma is a homological countepart of the classical Sperner lemma. It can also be seen as a homological counterpart of Musin's Sperner-type results for pseudomanifolds [15] and of Theorem 4.7 in the paper by Asada et al. [2]. We leave as an open question the existence of a proof based on a nerve theorem of some kind.We recall that a pseudomanifold is a simplicial complex that is pure, non-branching (each ridge is contained in exactly two facets), and strongly connected (the dual is connected). A colourful simplex in a simplicial complex whose vertices are partitioned into subsets V 0 , . . . , V m is a simplex with at most one vertex in each V i . Given a simplicial complex K and a subset U of its vertices, K[U] is the subcomplex induced by U, i.e. the simplicial complex whose simplices are exactly the simplices of K whose vertices are all in U.

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“…= 6. With Equalities (1) and (2), the fact that 4 does not divide 6 implies that (det •λ )(t) is nonzero. The conclusion is then identical.…”

Section: Proofsmentioning

confidence: 99%

Envy-free cake division without assuming the players prefer nonempty pieces

Meunier

Zerbib

2019

Isr. J. Math.

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Consider n players having preferences over the connected pieces of a cake, identified with the interval [0, 1]. A classical theorem, found independently by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it is possible to divide the cake into n connected pieces and assign these pieces to the players in an envy-free manner, i.e, such that no player strictly prefers a piece that has not been assigned to her. One of these conditions, considered as crucial, is that no player is happy with an empty piece. We prove that, even if this condition is not satisfied, it is still possible to get such a division when n is a prime number or is equal to 4. When n is at most 3, this has been previously proved by Erel Segal-Halevi, who conjectured that the result holds for any n. The main step in our proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and which is reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary.

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Fair Division and Generalizations of Sperner- and KKM-type Results

Cited by 23 publications

References 32 publications

Splitting Loops and Necklaces: Variants of the Square Peg Problem

Splitting Loops and Necklaces: Variants of the Square Peg Problem

Different versions of the nerve theorem and colourful simplices

Envy-free cake division without assuming the players prefer nonempty pieces

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