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Robust Tverberg and Colourful Carathéodory Results via Random Choice

Part of: General convexity Extremal combinatorics

Published online by Cambridge University Press:  19 December 2017

PABLO SOBERÓN
PABLO SOBERÓN*
Affiliation:
Mathematics Department, Northeastern University, Boston, MA 02445, USA (e-mail: p.soberonbravo@northeastern.edu)
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Abstract

We use the probabilistic method to obtain versions of the colourful Carathéodory theorem and Tverberg's theorem with tolerance.

In particular, we give bounds for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.

We prove a bound N = rt + O($\sqrt{t}$) for fixed r,d which is polynomial in each parameters. Our bounds extend to colourful versions of Tverberg's theorem, as well as Reay-type variations of this theorem.

MSC classification

Primary: 52A35: Helly-type theorems and geometric transversal theory
Secondary: 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 3 , May 2018 , pp. 427 - 440
Copyright
Copyright © Cambridge University Press 2017 

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References

[1] Ajtai, M., Chvátal, V., Newborn, M. M. and Szemerédi, E. (1982) Crossing-free subgraphs. In Theory and Practice of Combinatorics, Vol. 60 of North-Holland Mathematics Studies, North-Holland, pp. 9–12.Google Scholar
[2] Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, third edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.Google Scholar
[3] Arocha, J. L., Bárány, I., Bracho, J., Fabila, R. and Montejano, L. (2009) Very colorful theorems. Discrete Comput. Geom. 42 142–154.Google Scholar
[4] Asada, M., Chen, R., Frick, F., Huang, F., Polevy, M., Stoner, D., Tsang, L. H. and Wellner, Z. (2016) On Reay's relaxed Tverberg conjecture and generalizations of Conway's thrackle conjecture. arXiv:1608.04279Google Scholar
[5] Bárány, I. (1982) A generalization of Carathéodory's theorem. Discrete Math. 40 141–152.CrossRefGoogle Scholar
[6] Bárány, I. (2015) Tensors, colours, octahedra. In Geometry, Structure and Randomness in Combinatorics (Matoušek, J. et al., eds), Edizione della Normale, pp. 1–17.Google Scholar
[7] Bárány, I. Personal communication.Google Scholar
[8] Bárány, I. and Larman, D. G. (1992) A colored version of Tverberg's theorem. J. London Math. Soc. s2-45 314–320.Google Scholar
[9] Bárány, I. and Onn, S. (1997) Colourful linear programming and its relatives. Math. Oper. Res. 22 550–567.Google Scholar
[10] Blagojević, P. V. M., Frick, F. and Ziegler, G. M. (2014) Tverberg plus constraints. Bull. London Math. Soc. 46 953–967.Google Scholar
[11] Blagojević, P. V. M., Matschke, B. and Ziegler, G. M. (2011) Optimal bounds for a colorful Tverberg–Vrećica type problem. Adv. Math. 226 5198–5215.Google Scholar
[12] Blagojević, P. V. M., Matschke, B. and Ziegler, G. M. (2015) Optimal bounds for the colored Tverberg problem. J. Eur. Math. Soc. 17 739–754.Google Scholar
[13] Chazelle, B. and Friedman, J. (1990) A deterministic view of random sampling and its use in geometry. Combinatorica 10 229–249.CrossRefGoogle Scholar
[14] Clarkson, K. L. (1987) New applications of random sampling in computational geometry. Discrete Comput. Geom. 2 195–222.Google Scholar
[15] Clarkson, K. L., Eppstein, D., Miller, G. L., Sturtivant, C. and Teng, S.-H. (1996) Approximating center points with iterative Radon points. Internat. J. Comput. Geom. Appl. 6 357–377.Google Scholar
[16] Forge, D., Las Vergnas, M. and Schuchert, P. (2001) 10 points in dimension 4 not projectively equivalent to the vertices of a convex polytope. Europ. J. Combin. 22 705–708.CrossRefGoogle Scholar
[17] García-Colín, N. (2007) Applying Tverberg type theorems to geometric problems. PhD thesis, University College London.Google Scholar
[18] García-Colín, N. and Larman, D. (2015) Projective equivalences of k-neighbourly polytopes. Graphs Combin. 31 1403–1422.Google Scholar
[19] García-Colín, N., Raggi, M. and Roldán-Pensado, E. (2017) A note on the tolerant Tverberg theorem. Discrete Comput. Geom. 58, no. 3, 746–754.Google Scholar
[20] Haussler, D. and Welzl, E. (1987) ϵ-nets and simplex range queries. Discrete Comput. Geom. 2 127–151.Google Scholar
[21] Holmsen, A. F. (2016) The intersection of a matroid and an oriented matroid. Adv. Math. 290 1–14.Google Scholar
[22] Holmsen, A. F., Pach, J. and Tverberg, H. (2008) Points surrounding the origin. Combinatorica 28 633–644.CrossRefGoogle Scholar
[23] Larman, D. G. (1972) On sets projectively equivalent to the vertices of a convex polytope. Bull. London Math. Soc. 4 6–12.Google Scholar
[24] Liu, R. Y., Serfling, R. J. and Souvaine, D. L. (2006) Data Depth: Robust Multivariate Analysis, Computational Geometry, and Applications, Vol. 72 of DIMAC Series in Discrete Mathematics and Theoretical Computer Science, AMS.Google Scholar
[25] Matoušek, J. (2002) Lectures on Discrete Geometry, Vol. 212 of Graduate Texts in Mathematics, Springer.Google Scholar
[26] Miller, G. L. and Sheehy, D. R. (2009) Approximate center points with proofs. In SCG '09: Twenty-Fifth Annual Symposium on Computational Geometry, ACM, pp. 153–158.Google Scholar
[27] Montejano, L. and Oliveros, D. (2011) Tolerance in Helly-type theorems. Discrete Comput. Geom. 45 348–357.Google Scholar
[28] Mulzer, W. and Stein, Y. (2013) Algorithms for tolerated Tverberg partitions. In ISAAC 2013: International Symposium on Algorithms and Computation, Springer, pp. 295–305.Google Scholar
[29] Perles, M. A. and Sigron, M. (2016) Some variations on Tverberg's theorem. Israel J. Math. 216 957–972.Google Scholar
[30] Reay, J. R. (1979) Several generalizations of Tverberg's theorem. Israel J. Math. 34 238–244.Google Scholar
[31] Rolnick, D. and Soberón, P. (2016) Algorithms for Tverberg's theorem via centerpoint theorems. arXiv:1601.03083v2Google Scholar
[32] Sarkaria, K. S. (1992) Tverberg's theorem via number fields. Israel J. Math. 79 317–320.Google Scholar
[33] Soberón, P. (2015) Equal coefficients and tolerance in coloured Tverberg partitions. Combinatorica 35 235–252.Google Scholar
[34] Soberón, P. and Strausz, R. (2012) A generalisation of Tverberg's theorem. Discrete Comput. Geom. 47 455–460.CrossRefGoogle Scholar
[35] Székely, L. A. (1997) Crossing numbers and hard Erdős problems in discrete geometry. Combin. Probab. Comput. 6 353–358.Google Scholar
[36] Tukey, J. W. (1975) Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians, Vol. 2, Canadian Mathematical Congress, pp. 523–531.Google Scholar
[37] Tverberg, H. (1966) A generalization of Radon's theorem. J. London Math. Soc. 41 123–128.Google Scholar