The Radon-split and the Helly-core of a point configuration

Onn, Shmuel

doi:10.1007/s00022-001-8577-x

Cited by 5 publications

(5 citation statements)

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“…Associate with every edge {v i , v j } in G its characteristic vector in R d , giving a configuration Conf(G) of 3n/2 points in (n − 1)-dimensional space. In [Onn01] and also in personal communication (2011), Onn observed that the existence of a Tverberg 3-partition (or even of a Reay (3,2)-partition; see Section 6.2) is equivalent to a 3-edge coloring for G, and he concluded that deciding if a configuration of 3(d + 1)/2 points in R d (d an odd integer) admits a Tverberg partition into three parts is NP-complete.…”

Section: Radon Partitions and Radon Points For Configurations Based O...mentioning

confidence: 99%

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2022

Bull. Amer. Math. Soc.

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In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals. Helly, Carathéodory, and Radon theoremsIn this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg.Helly's theorem [Hel23] asserts that for a family {K 1 , K 2 , . . . , K n } of convex sets in R d , where n ≥ d + 1, if every d + 1 of the sets have a point in common, then all of the sets have a point in common. The closely related Carathéodory theorem [Car07] states that forThe more general colorful Carathéodory theorem [Bár82] says the following. Let S 1 , S 2 , . . . , S d+1 be d + 1 sets (or colors if you wish) in R d . Suppose that x ∈ d+1 i=1 conv S i . Then there is a transversal T = {x 1 , . . . , x d+1 } of the system S 1 , . . . , S d+1 , meaning that x 1 ∈ S 1 , x 2 ∈ S 2 , . . . , x d+1 ∈ S d+1 such that x ∈ conv T . A transversal is also called a rainbow set when S 1 , . . . , S d+1 are considered as colors. The uncolored version, that is, when S 1 = S 2 = • • • = S d+1 , is the classic result of Carathéodory. There is a closely related colorful version of Helly's theorem due to Lovász that appeared in [Bár82].Tverberg's theorem [Tve66] states the following. Let x 1 , x 2 , . . . , x m be points in R d with m ≥ (r − 1)(d + 1) + 1. Then there is a partition S 1 , S 2 , . . . , S r of {1, 2, . . . , m} such that r j=1 conv {x i : i ∈ S j } = ∅. This was a conjecture by Birch, who also proved the planar case in a slightly different form. The bound of (r −1)(d+1)+1 in the theorem is sharp as can easily be seen from the configuration of points in a sufficiently general position.The case r = 2 is Radon's theorem [Rad21], another classic from 1921, which was used by Radon to prove Helly's theorem. Helly's original proof (published later) was based on a separation argument. Sarkaria [Sar92] gave a simple proof of Tverberg's theorem based on the colorful Carathéodory theorem.

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Section: Radon Partitions and Radon Points For Configurations Based O...mentioning

confidence: 99%

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2022

Bull. Amer. Math. Soc.

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“…Equality was conjectured [28,31], and actually holds when d = 2 or m = 1. However, Avis found a counterexample for n = 9, d = 3 and m = 3 [1], and Onn provided a systematic construction for counterexamples [26].…”

Section: Theorem Bmentioning

confidence: 99%

The degree of point configurations: Ehrhart theory, Tverberg points and almost neighborly polytopes

Nill

Padrol

2015

European Journal of Combinatorics

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The degree of a point configuration is defined as the maximal codimension of its interior faces. This concept is motivated from a corresponding Ehrhart-theoretic notion for lattice polytopes and is related to neighborly polytopes and the generalized lower bound theorem and, by Gale duality, to Tverberg theory.The main results of this paper are a complete classification of point configurations of degree 1, as well as a structure result on point configurations whose degree is less than a third of the dimension. Statements and proofs involve the novel notion of a weak Cayley decomposition, and imply that the m-core of a set S of n points in R r is contained in the set of Tverberg points of order (3m − 2(n − r)) of S. 1 arXiv:1209.5712v2 [math.CO]

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“…Equality was conjectured [Rea82,Sie82], and actually holds when r = 2 or κ = 1. However, Avis found a counterexample for n = 9, r = 3 and κ = 3 [Avi93], and Onn provided a systematic construction for counterexamples [Onn01].…”

Section: Tverberg's Theorymentioning

confidence: 99%

Neighborly and almost neighborly configurations, and their duals

Padrol Sureda

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This thesis presents new applications of Gale duality to the study of polytopes with extremal combinatorial properties. It consists in two parts. The first one is devoted to the construction of neighborly polytopes and oriented matroids. The second part concerns the degree of point configurations, a combinatorial invariant closely related to neighborliness. A d-dimensional polytope P is called neighborly if every subset of at most d/2 vertices of P forms a face. In 1982, Ido Shemer presented a technique to construct neighborly polytopes, which he named the "Sewing construction". With it he could prove that the number of neighborly polytopes in dimension d with n vertices grows superexponentially with n. One of the contributions of this thesis is the analysis of the sewing construction from the point of view of lexicographic extensions. This allows us to present a technique that we call the "Extended Sewing construction", that generalizes it in several aspects and simplifies its proof. We also present a second generalization that we call the "Gale Sewing construction". This construction exploits Gale duality an is based on lexicographic extensions of the duals of neighborly polytopes and oriented matroids. Thanks to this technique we obtain one of the main results of this thesis: a lower bound of ((r+d)^(((r+d)/2)^2)/(r^((r/2)^2)d^((d/2)^2)e^(3rd/4)) for the number of combinatorial types of neighborly polytopes of even dimension d and r+d+1 vertices. This result not only improves Shemer's bound, but it also improves the current best bounds for the number of polytopes. The combination of both new techniques also allows us to construct many non-realizable neighborly oriented matroids. The degree of a point configuration is the maximal codimension of its interior faces. In particular, a simplicial polytope is neighborly if and only if the degree of its set of vertices is [(d+1)/2]. For this reason, d-dimensional configurations of degree k are also known as "(d-k)-almost neighborly". The second part of the thesis presents various results on the combinatorial structure of point configurations whose degree is small compared to their dimension; specifically, those whose degree is smaller than [(d+1)/2], the degree of neighborly polytopes. The study of this problem comes motivated by Ehrhart theory, where a notion equivalent to the degree - for lattice polytopes - has been widely studied during the last years. In addition, the study of the degree is also related to the "generalized lower bound theorem" for simplicial polytopes, with Cayley polytopes and with Tverberg theory. Among other results, we present a complete combinatorial classification for point configurations of degree 1. Moreover, we show combinatorial restrictions in terms of the novel concept of "weak Cayley configuration" for configurations whose degree is smaller than a third of the dimension. We also introduce the notion of "codegree decomposition" and conjecture that any configuration whose degree is smaller than half the dimension admits a non-trivial codegree decomposition. For this conjecture, we show various motivations and we prove some particular cases.

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The Radon-split and the Helly-core of a point configuration

Cited by 5 publications

References 8 publications

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The degree of point configurations: Ehrhart theory, Tverberg points and almost neighborly polytopes

Neighborly and almost neighborly configurations, and their duals

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