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<j≤d+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.
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References
Arocha, J.L., Bárány, I., Bracho, J., Fabila, R., Montejano, L.: Very colorful theorems. Discret. Comput. Geom. 42, 142–154 (2009)
Bárány, I.: A generalization of Carathéodory’s theorem. Discret. Math. 40, 141–152 (1982)
Bárány, I., Matoušek, J.: Quadratically many colorful simplices. SIAM J. Discret. Math. 21, 191–198 (2007)
Bárány, I., Onn, S.: Colourful linear programming and its relatives. Math. Oper. Res. 22, 550–567 (1997)
Custard, G., Deza, A., Stephen, T., Xie, F.: Small octahedral systems. In: Proceedings of the 23rd Canadian Conference on Computational Geometry (CCCG’11), Toronto (2011)
Deza, A., Huang, S., Stephen, T., Terlaky, T.: Colourful simplicial depth. Discret. Comput. Geom. 35, 597–604 (2006)
Deza, A., Huang, S., Stephen, T., Terlaky, T.: The colourful feasibility problem. Discret. Appl. Math. 156, 2166–2177 (2008)
Deza, A., Stephen, T., Xie, F.: More colourful simplices. Discret. Comput. Geom. 45, 272–278 (2011)
Edmonds, J., Sanità, L.: On finding another room-partitioning of the vertices. Electron. Notes Discret. Math. 36, 1257–1264 (2010)
Grigni, M.: A Sperner lemma complete for PPA. Inf. Process. Lett. 77, 255–259 (2001)
Holmsen, A.F., Pach, J., Tverberg, H.: Points surrounding the origin. Combinatorica 28, 633–644 (2008)
Papadimitriou, C.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48, 498–532 (1994)
Stephen, T., Thomas, H.: A quadratic lower bound for colourful simplicial depth. J. Comb. Optim. 16, 324–327 (2008)
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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Meunier, F., Deza, A. (2013). A Further Generalization of the Colourful Carathéodory Theorem. In: Bezdek, K., Deza, A., Ye, Y. (eds) Discrete Geometry and Optimization. Fields Institute Communications, vol 69. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00200-2_11
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DOI: https://doi.org/10.1007/978-3-319-00200-2_11
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