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Positive-fraction intersection results and variations of weak epsilon-nets

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Abstract

Given a finite set X of points in \(R^n\) and a family F of sets generated by the pairs of points of X, we determine volumetric and structural conditions for the sets that allow us to guarantee the existence of a positive-fraction subfamily \(F'\) of F for which the sets have non-empty intersection. This allows us to show the existence of weak epsilon-nets for these families. We also prove a topological variation of the existence of weak epsilon-nets for convex sets.

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References

  1. Alon, N., Bárány, I., Füredi, Z., Kleitman, D.J.: Point selections and weak \(\varepsilon \)-nets for convex hulls. Combin. Probab. Comput. 1(03), 189–200 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bárány, I.: A generalization of Carathéodory’s theorem. Discrete Math. 40(2–3), 141–152 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bárány, I.: An extension of the Erdős–Szekeres theorem on large angles. Combinatorica 7(2), 161–169 (1987)

    Article  MathSciNet  Google Scholar 

  4. Bárány, I., Füredi, Z.: Computing the volume is difficult. Discrete Comput. Geom. 2(1), 319–326 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bárány, I., Lehel, J.: Covering with Euclidean boxes. Eur. J. Combin. 8(2), 113–119 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bárány, I., Shlosman, S.B., Szücs, A.: On a topological generalization of a theorem of Tverberg. J. Lond. Math. Soc. 2(1), 158–164 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boros, E., Füredi, Z.: The number of triangles covering the center of an n-set. Geom. Dedicata 17(1), 69–77 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bukh, B.: A point in many triangles. Electron. J. Combin. 13(1), Note 10, 3 pp (2006)

  9. Bukh, B., Matoušek, J., Nivasch, G.: Stabbing simplices by points and flats. Discrete Comput. Geom. 43(2), 321–338 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bukh, B., Matoušek, J., Nivasch, G.: Lower bounds for weak epsilon-nets and stair-convexity. Isr. J. Math. 182(1), 199–228 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L., Sharir, M., Welzl, E.: Improved bounds on weak \(\varepsilon \)-nets for convex sets. Discrete Comput. Geom. 13(1), 1–15 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fox, J., Gromov, M., Lafforgue, V., Naor, A., Pach, J.: Overlap properties of geometric expanders. J. Reine Angew. Math. 671, 49–83 (2012)

  13. Gromov, M.: Singularities, expanders and topology of maps. Part 2: from combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal. 20(2), 416–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jiang, Z.: A slight improvement to the colored bárány’s theorem. Electron. J. Combin. 21(4), 1–8 (2014)

    MATH  Google Scholar 

  15. Karasev, R.N.: A simpler proof of the Boros–Füredi–Bárány–Pach–Gromov theorem. Discrete Comput. Geom. 47(3), 492–495 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Katchalski, M., Liu, A.: A problem of geometry in \(R^n\). Proc. Am. Math. Soc. 75(2), 284–288 (1979)

    MATH  Google Scholar 

  17. Král, D., Mach, L., Sereni, J.-S.: A new lower bound based on Gromov’s method of selecting heavily covered points. Discrete Comput. Geom. 48(2), 487–498 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Matoušek, J.: Using the Borsuk–Ulam Theorem. Universitext. Lectures on topological methods in combinatorics and geometry. Springer, Berlin (2003). Written in cooperation with Anders Björner and Günter M. Ziegler

  19. Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)

  20. Matoušek, J., Wagner, U.: New constructions of weak epsilon-nets. Discrete Comput. Geom. 32(2), 195–206 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Matoušek, J., Wagner, U.: On Gromov’s method of selecting heavily covered points. Discrete Comput. Geom. 52(1), 1–33 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pach, J.: A Tverberg-type result on multicolored simplices. Comput. Geom. 10(2), 169–178 (1998)

    MathSciNet  MATH  Google Scholar 

  23. Rado, R.: A theorem on general measure. J. Lond. Math. Soc. 21, 291–300 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rolnick, D., Soberón, P.: Quantitative \((p, q)\) theorems in combinatorial geometry (2015). arXiv:1504.01642 [math.MG]

  25. Tverberg, H.: A generalization of Radon’s theorem. J. Lond. Math. Soc 41(1), 123–128 (1966)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Rényi Mathematical Institute, Budapest, Hungary and Yandex LLC, Moscow, Russia

    Alexander Magazinov

  2. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Pablo Soberón

Authors
  1. Alexander Magazinov
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  2. Pablo Soberón
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Corresponding author

Correspondence to Pablo Soberón.

Additional information

Communicated by A. Constantin.

A. Magazinov was Supported by ERC Advanced Research Grant No. 267165 (DISCONV).

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Magazinov, A., Soberón, P. Positive-fraction intersection results and variations of weak epsilon-nets. Monatsh Math 183, 165–176 (2017). https://doi.org/10.1007/s00605-016-0892-2

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  • DOI: https://doi.org/10.1007/s00605-016-0892-2

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