A generalization of carathéodory's theorem

Bárány, Imre

doi:10.1016/0012-365x(82)90115-7

Cited by 264 publications

(330 citation statements)

References 1 publication

Supporting

Mentioning

322

Contrasting

Unclassified

Order By: Relevance

“…We remark that the D-depth of any point is at least one. This follows from the result in [1] that every point in a colourful configuration with 0 in its core is among the generators of at least one colourful simplex containing 0.…”

Section: Preliminariesmentioning

confidence: 88%

A quadratic lower bound for colourful simplicial depth

Stephen

Thomas

2008

J Comb Optim

View full text Add to dashboard Cite

show abstract

Section: Preliminariesmentioning

confidence: 88%

A quadratic lower bound for colourful simplicial depth

Stephen

Thomas

2008

J Comb Optim

View full text Add to dashboard Cite

show abstract

“…This is a particular case of Bárány [3] the argument being as follows. If conv s(g α e α ) : g α ∈ G, 1 ≤ α ≤ N + 1 is at a distance δ > 0 from 0 ∈ E, then its nearest point P is contained in the hyperplane H normal to 0P and out of the points s(g α e α ) we can choose ≤ N which all lie on H and are such that P is in their convex hull.…”

Section: Proof Of Theorems Of Tverberg and Báránymentioning

confidence: 99%

Tverberg partitions and Borsuk–Ulam theorems

Sarkaria¹

2000

Pacific J. Math.

View full text Add to dashboard Cite

An N -dimensional real representation E of a finite group G is said to have the "Borsuk-Ulam Property" if any continuous G-map from the (N + 1)-fold join of G (an N -complex equipped with the diagonal G-action) to E has a zero. This happens iff the "Van Kampen characteristic class" of E is nonzero, so using standard computations one can explicitly characterize representations having the B-U property. As an application we obtain the "continuous" Tverberg theorem for all prime powers q, i.e., that some q disjoint faces of a (q − 1)(d + 1)-dimensional simplex must intersect under any continuous map from it into affine d-space. The "classical" Tverberg, which makes the same assertion for all linear maps, but for all q, is explained in our set-up by the fact that any representation E has the analogously defined "linear B-U property" iff it does not contain the trivial representation.

show abstract

“…problems whose decision version has always a yes answer. The geometric algorithms introduced by Bárány [Bár82] and Bárány and Onn [BO97] and other methods to tackle the colourful feasibility problem, such as multi-update modifications, are studied and benchmarked in [DHST08]. The complexity of this challenging problem, i.e.…”

Section: Colourful Carathéodory Theoremsmentioning

confidence: 99%

A Further Generalization of the Colourful Carathéodory Theorem

Meunier

Deza

2013

Discrete Geometry and Optimization

View full text Add to dashboard Cite

Given d +1 sets, or colours, S 1 , S 2 , . . . ,The convex hull of a colourful set S is called a colourful simplex. Bárány's colourful Carathéodory theorem asserts that if the origin 0 is contained in the convex hull of S i for i = 1, . . . , d + 1, then there exists a colourful simplex containing 0. The sufficient condition for the existence of a colourful simplex containing 0 was generalized to 0 being contained in the convex hull of S i ∪S j for 1 ≤ i < j ≤ d + 1 by Arocha et al. and by Holmsen et al. We further generalize the sufficient condition and obtain new colourful Carathéodory theorems. We also give an algorithm to find a colourful simplex containing 0 under the generalized condition. In the plane an alternative, and more general, proof using graphs is given. In addition, we observe that any condition implying the existence of a colourful simplex containing 0 actually implies the existence of min i |S i | such simplices.

show abstract

A generalization of carathéodory's theorem

Cited by 264 publications

References 1 publication

A quadratic lower bound for colourful simplicial depth

A quadratic lower bound for colourful simplicial depth

Tverberg partitions and Borsuk–Ulam theorems

A Further Generalization of the Colourful Carathéodory Theorem

Contact Info

Product

Resources

About