A Generalisation of Tverberg’s Theorem

Soberón, Pablo; Strausz, Ricardo

doi:10.1007/s00454-011-9379-z

Cited by 22 publications

(27 citation statements)

References 10 publications

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“…For small t, the bound above falls short of an earlier bound N ≤ (r − 1)(d + 1)(t + 1) + 1 [SS12]. The only case when the optimal number is known is d = 1, N (r, t, 1) = rt + 2r − 1, by Mulzer and Stein [MS14], who studied algorithmic versions of the problem.…”

Section: Tverberg With Tolerancementioning

confidence: 86%

Tverberg’s theorem is 50 years old: A survey

Bárány

Soberón

2018

Bull. Amer. Math. Soc.

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This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. r 1 (z − y j ) = 0. Note that z = y j is possible but cannot hold for all j since µ > 0.Define Y j ⊂ X j for j = 1, . . . , r via y j ∈ relint conv Y j . We claim that r 1 aff Y j = ∅. Otherwise there is a point v ∈ r 1 aff Y j . Let ·, · denote the standard scalar product, so x, x = x 2 , for instance. Then z − v, z − y j > 0 if y i = z (because y j is the closest point to z in conv Y j ) Imre Bárány

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Section: Tverberg With Tolerancementioning

confidence: 86%

Tverberg’s theorem is 50 years old: A survey

Bárány

Soberón

2018

Bull. Amer. Math. Soc.

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“…The theorem above implies Tverberg's Theorem with tolerance from [19] in the same way that Theorem 1.1 implies Tverberg's Theorem. The proofs of Theorems 1.1 and 1.2 are given in section 3.…”

Section: Introductionmentioning

confidence: 60%

“…We say that a property P of points in R d is true in F with tolerance r if P (F \C) is true for all C ⊂ F of size r. For example, P can be "0 is in the convex hull of F ". There are now versions of the classical Helly, Carathéodory and Tverberg theorems with a tolerance condition [15], [19]. We show that the previous theorem also has a version with tolerance, but in this case we do not know that the number of colour classes is optimal.…”

Section: Introductionmentioning

confidence: 90%

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Equal coefficients and tolerance in coloured tverberg partitions

Soberón

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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The coloured Tverberg theorem was conjectured by Bárány, Lovász and Füredi [2] and asks whether for any d + 1 sets (considered as colour classes) of k points each in R d there is a partition of them into k colourful sets whose convex hulls intersect. This is known when d = 1, 2 [3] or k + 1 is prime [5]. In this paper we show that (k − 1)d + 1 colour classes are necessary and sufficient if the coefficients in the convex combination in the colourful sets are required to be the same in each class. We also examine what happens if we want the convex hulls of the colourful sets to intersect even if we remove any r of the colour classes. Namely, if we have (r + 1)(k − 1)d + 1 colour classes of k point each, there is a partition of them into k colourful sets such that they intersect using the same coefficients regardless of which r colour classes are removed. We also investigate the relation of the case k = 2 and the Gale transform. We then show applications of these results to purely combinatorial problems.

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“…In the notation of Theorem 1.3, Theorem 1.4 says that if ε > 1−1/r, then m(ε, r) = 1. Improved bounds for small values of t can be found in [SS12,MS14].…”

Section: Introductionmentioning

confidence: 99%