Advertisement

Tverberg Partitions as Weak Epsilon-Nets

  • Pablo SoberónEmail author
Article

Mathematics Subject Classification (2010)

52A35 05D40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Alon, I. Bárány, Z. Füredi and D. J. Kleitman: Point selections and weak ɛ-nets for convex hulls, Combin. Probab. Comput. 1 (1992), 189–200.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    N. Alon and J. H. Spencer: The probabilistic method, fourth ed., Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2016.Google Scholar
  3. [3]
    K. Azuma: Weighted sums of certain dependent random variables, Tôhoku Math. J. 19 (1967), 357–367.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    I. Bárány: A generalization of Carathéodory’s theorem, Discrete Math. 40 (1982), 141–152.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    I. Bárány: Tensors, colours, octahedra, Geometry, structure and randomness in com-binatorics, CRM Series, vol. 18, Ed. Norm., Pisa, 2015, 1–17.Google Scholar
  6. [6]
    P. V. M. Blagojević, F. Frick and G. M. Ziegler: Tverberg plus constraints, Bull. Lond. Math. Soc. 46 (2014), 953–967.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    I. Bárány and D. G. Larman: A Colored Version of Tverberg’s Theorem, J. London Math. Soc. s2-45 (1992), 314–320.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    B. Bukh, J. Matoušek and G. Nivasch: Lower bounds for weak epsilon-nets and stair-convexity, Israel J. Math. 182 (2011), 199–228.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    P.V.M. Blagojević, B. Matschke and G. M. Ziegler: Optimal bounds for a colorful Tverberg-Vrećica type problem, Adv. Math. 226 (2011), 5198–5215.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    P. V. M. Blagojević, B. Matschke and G. M. Ziegler: Optimal bounds for the colored Tverberg problem, J. Eur. Math. Soc. (JEMS) 17 (2015), 739–754.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    I. Bárány and S. Onn: Colourful linear programming and its relatives, Math. Oper. Res. 22 (1997), 550–567.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    I. Bárány and P. Soberón: Tverberg plus minus, Discrete Comput. Geom. 60 (2018), 588–598.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    I. Bárány and P. Soberón: Tverberg’s theorem is 50 years old: a survey, Bull. Amer. Math. Soc. 55 (2018), 459–492.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    B. Chazelle, H. Edelsbrunner, M. Grigni, L. Guibas, M. Sharir and E. Welzl: Improved bounds on weak e-nets for convex sets, Discrete Comput. Geom. 13 (1995), 1–15.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    K. L. Clarkson, D. Eppstein, G. L. Miller, C. Sturtivant and S.-H. Teng: Approximating center points with iterative Radon points, Internat. J. Comput. Geom. Appl. 6 (1996), 357–377, ACM Symposium on Computational Geometry (San Diego, CA, 1993).MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    N. García-Colín, M. Raggi and E. Roldán-Pensado: A note on the tolerant Tverberg theorem, Discrete Comput. Geom. 58 (2017), 746–754.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    W. Hoeffding: Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    D. G. Larman: On Sets Projectively Equivalent to the Vertices of a Convex Polytope, Bull. Lond. Math. Soc. 4 (1972), 6–12.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    J. Matoušek: Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002.CrossRefGoogle Scholar
  20. [20]
    W. Mulzer and Y. Stein: Algorithms for tolerant Tverberg partitions, Internat. J. Comput. Geom. Appl. 24 (2014), 261–273.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J. Matoušek and U. Wagner: New constructions of weak e-nets, Discrete Comput. Geom. 32 (2004), 195–206.MathSciNetzbMATHGoogle Scholar
  22. [22]
    K. S. Sarkaria: Tverberg’s theorem via number fields, Israel J. Math. 79 (1992), 317–320.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    P. Soberón: Equal coefficients and tolerance in coloured tverberg partitions, Combinatorica 35 (2015), 235–252.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    P. Soberón: Robust Tverberg and Colourful Carathéodory results via Random Choice, Combinatorics, Probability and Computing 27 (2018), 427–440.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    P. Soberón and R. Strausz: A generalisation of Tverberg’s theorem, Discrete Comput. Geom. 47 (2012), 455–460.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    H. Tverberg: A generalization of Radon’s theorem, J. London Math. Soc. 41 (1966), 123–128.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentBaruch College, CUNYNew YorkUSA

Personalised recommendations