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Article Dans Une Revue Discrete Mathematics Année : 2016

An Optimal Generalization of the Colorful Carathéodory Theorem

Nabil Mustafa
Saurabh Ray
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NYU

Résumé

The Colorful Carathéodory theorem by Bárány (1982) states that given d + 1 sets of points in R d , the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d + 1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the Colorful Carathéodory theorem: given + 1 sets of points in R d and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a rainbow simplex intersecting C.

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hal-01233467 , version 1 (25-11-2015)

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Nabil Mustafa, Saurabh Ray. An Optimal Generalization of the Colorful Carathéodory Theorem. Discrete Mathematics, 2016, ⟨10.1016/j.disc.2015.11.019⟩. ⟨hal-01233467⟩
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