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A Further Generalization of the Colourful Carathéodory Theorem

Chapter
Part of the Fields Institute Communications book series (FIC, volume 69)

Abstract

Given d+1 sets, or colours, \(\mathbf{S}_{1},\mathbf{S}_{2},\ldots,\mathbf{S}_{d+1}\) of points in \({\mathbb{R}}^{d}\), a colourful set is a set \(S \subseteq \bigcup _{i}\mathbf{S}_{i}\) such that \(\vert S \cap \mathbf{S}_{i}\vert \leq 1\) for \(i = 1,\ldots,d + 1\). 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 \(i = 1,\ldots,d + 1\), 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 \(\mathbf{S}_{i} \cup \mathbf{S}_{j}\) for 1≤i<jd+1 by Arocha etal. and by Holmsen etal. 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.

Key words

Colourful Carathéodory theorem Colourful simplicial depth Discrete geometry 

Subject Classifications

52C45 52A35 

Notes

Acknowledgements

This work was supported by grants from NSERC, MITACS, and Fondation Sciences Mathématiques de Paris, and by the Canada Research Chairs program. We are grateful to Sylvain Sorin and Michel Pocchiola for providing the environment that nurtured this work from the beginning.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Université Paris Est, CERMICSMarne-la-Vallée Cedex 2France
  2. 2.Advanced Optimization Laboratory, Department of Computing and SoftwareMcMaster UniversityHamiltonCanada
  3. 3.Equipe Combinatoire et OptimisationUniversité Pierre et Marie CurieParisFrance

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