Helly-type problems

Bárány, Imre; Kalai, Gil

doi:10.1090/bull/1753

Cited by 11 publications

(6 citation statements)

References 97 publications

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“…It is known that any group of N ⋅ ( d + 1) points in a d -dimensional space can be partitioned into subsets of size d + 1 so that there is a point that belongs to the convex hull of all these subsets. This result was first proven in Birch (1959) for dimension d = 2, then in Frick and Soberón (2020) for all dimensions d ; see also Bárány (2022) for the general overview of this and related results. In our case, we have a space of dimension d = n − 1, so the above result indeed implies the first part of our proposition.…”

Section: Proofmentioning

confidence: 71%

How to reach a joint decision with the smallest need for compromise

Holguin

Kosheleva

2023

AJEB

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PurposeUsually, people's interests do not match perfectly. So when several people need to make a joint decision, they need to compromise. The more people one has to coordinate the decision with, the fewer chances that each person's preferences will be properly taken into account. Therefore, when a large group of people need to make a decision, it is desirable to make sure that this decision can be reached by dividing all the people into small-size groups so that this decision can reach a compromise between the members of each group. The study's objective is to analyze when such a compromise is possible.Design/methodology/approachIn this paper, the authors use a recent mathematical result about convex sets to analyze this problem and to come up with an optimal size of such groups.FindingsThe authors find the smallest group size for which a joint decision is possible. Specifically, the authors show that in situations where each alternative is characterized by n quantities, it is possible to have a joint decision if the participants are divided into groups of size n -- and, in general, no such decision is possible if the participants are divided into groups of size n -- 1.Originality/valueThe main novelty of this paper is that, first, it formulates the problem, which, to the best of the authors’ knowledge, was never formulated in this way before, and, second, that it provides a solution to this problem.

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Section: Proofmentioning

confidence: 71%

How to reach a joint decision with the smallest need for compromise

Holguin

Kosheleva

2023

AJEB

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“…The problem of finding the optimal values of δ(d) and v(d) has enjoyed special interest in recent years (see e.g. the excellent survey article [BK21]). In [BKP82] (see also [BKP84]) the authors proved that v(d) ≤ d 2d 2 and δ(d) ≤ d 2d , and they conjectured that v(d) ≈ d c 1 d and δ(d) ≈ c 2 d 1/2 for some positive constants c 1 , c 2 > 0.…”

Section: History and Resultsmentioning

confidence: 99%

Quantitative Helly-Type Theorems via Sparse Approximation

Almendra-Hernández

Ambrus

Kendall

2022

Discrete Comput Geom

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We prove the following sparse approximation result for polytopes. Assume that Q is a polytope in John's position. Then there exist at most 2d vertices ofAs a consequence, we retrieve the best bound for the quantitative Helly-type result for the volume, achieved by Brazitikos, and improve on the strongest bound for the quantitative Helly-type theorem for the diameter, shown by Ivanov and Naszódi: We prove that given a finite family F of convex bodies in R d with intersection K, we may select at most 2d members of F such that their intersection has volume at most (cd) 3d/2 vol K, and it has diameter at most 2d 2 diam K, for some absolute constant c > 0.

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“…In Bárány and Kalai (2022) they survey recent developments related to classical convexity results. In particular, following Moran and Yehudayoff (2020), given a hypergraph (with bounded VC-dimension), whose hyperedges represent 'halfspaces', they define MY-convexity where the convex sets are the intersections of these halfspaces.…”

Section: P-convex Hulls Of Pseudolinesmentioning

confidence: 99%

A new discrete theory of pseudoconvexity

Keszegh¹

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were defined in a purely combinatorial way. This led to extensions of earlier results about points and halfplanes to pseudohalfplanes, including polychromatic colorings and discrete Helly-type theorems about pseudohalfplanes. Here we continue this line of research and introduce the notion of convex sets of such pseudohalfplane hypergraphs. In this context we prove several results corresponding to classical results about convexity, namely Helly's Theorem, Carath\'eodory's Theorem, Kirchberger's Theorem, Separation Theorem, Radon's Theorem and the Cup-Cap Theorem. These results imply the respective results about pseudoconvex sets in the plane defined using pseudohalfplanes. It turns out that most of our results can be also proved using oriented matroids and topological affine planes (TAPs) but our approach is different from both of them. Compared to oriented matroids, our theory is based on a linear ordering of the vertex set which makes our definitions and proofs quite different and perhaps more elementary. Compared to TAPs, which are continuous objects, our proofs are purely combinatorial and again quite different in flavor. Altogether, we believe that our new approach can further our understanding of these fundamental convexity results.

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Helly-type problems

Cited by 11 publications

References 97 publications

How to reach a joint decision with the smallest need for compromise

How to reach a joint decision with the smallest need for compromise

Quantitative Helly-Type Theorems via Sparse Approximation

A new discrete theory of pseudoconvexity

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