Advertisement

Functional Covering Numbers

  • Shiri Artstein-AvidanEmail author
  • Boaz A. Slomka
Article
  • 9 Downloads

Abstract

We define covering and separation numbers for functions. We investigate their properties, and show that for some classes of functions there is exact equality between separation and covering numbers. We provide analogues for various geometric inequalities on covering numbers, such as volume bounds, bounds connected with Hadwiger’s conjecture, and inequalities about M-positions for geometric log-concave functions. In particular we get strong versions of M-positions for geometric log-concave functions.

Keywords

Covering numbers Functionalization of geometry Log-concave functions Duality Volume bounds M-position 

Mathematics Subject Classification

52C17 52A23 46A20 

Notes

Acknowledgements

We thank Daniel Rosen for his valuable remarks for fruitful discussions. We also thank the anonymous referees for helpful remarks. This publication is a part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 770127). The first named author was supported by ISF Grant Number 665/15.

References

  1. 1.
    Alonso-Gutiérrez, D., Merino, B.González, Jiménez, C.H., Villa, R.: Rogers-Shephard inequality for log-concave functions. J. Funct. Anal. 271(11), 3269–3299 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Artstein-Avidan, S., Giannopoulos, A., Milman, V.D.: Asymptotic geometric analysis. Part I, Mathematical Surveys and Monographs, vol. 202, American Mathematical Society, Providence, RI (2015)Google Scholar
  3. 3.
    Artstein-Avidan, S., Klartag, B., Milman, V.: The Santaló point of a function, and a functional form of the Santaló inequality. Mathematika 5(1), 33–48 (2004)CrossRefGoogle Scholar
  4. 4.
    Artstein-Avidan, S., Milman, V.: A characterization of the support map. Adv. Math. 223(1), 379–391 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Artstein-Avidan, S., Milman, V.: Hidden structures in the class of convex functions and a new duality transform. J. Eur. Math. Soc. 13(4), 975–1004 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Artstein-Avidan, S., Raz, O.: Weighted covering numbers of convex sets. Adv. Math. 227(1), 730–744 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Artstein-Avidan, S., Slomka, B.A.: On weighted covering numbers and the Levi-Hadwiger conjecture. Israel J. Math. 209(1), 125–155 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ball, K.M.: PhD dissertation, CambridgeGoogle Scholar
  9. 9.
    Barvinok, A.: A course in convexity, Graduate Studies in Mathematics, vol. 54, American Mathematical Society, Providence, RI (2002)Google Scholar
  10. 10.
    Bobkov, S., Madiman, M.: Reverse brunn-minkowski and reverse entropy power inequalities for convex measures. J. Funct. Anal. 262(7), 3309–3339 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bourgain, J., Milman, V.: New volume ratio properties for convex symmetric bodies in \({{\bf R}}^n\). Invent. Math. 88(2), 319–340 (1987)Google Scholar
  12. 12.
    Fradelizi, M., Meyer, M.: Increasing functions and inverse Santaló inequality for unconditional functions. Positivity 12(3), 407–420 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gohberg, I., Markus, A.: A problem on covering of convex figures by similar figures (in Russian). Izv. Mold. Fil. Akad. Nauk SSSR 10(76), 87–90 (1960)Google Scholar
  14. 14.
    Hadwiger, H.: Ungelöstes Probleme Nr. 20. Elem. Math. 12(6), 121 (1957)MathSciNetGoogle Scholar
  15. 15.
    Klartag, B., Milman, V.D.: Geometry of log-concave functions and measures. Geom. Dedicata 112(1), 169–182 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    König, H., Milman, V.D.: On the covering numbers of convex bodies, Geometrical Aspects of Functional Analysis (Joram Lindenstrauss and Vitali D. Milman, eds.), Lecture Notes in Mathematics, vol. 1267, Springer, Berlin, 1987, pp. 82–95 (English)Google Scholar
  17. 17.
    Levi, F.W.: Überdeckung eines Eibereiches durch Parallelverschiebung seines offenen Kerns. Arch. Math. (Basel) 6, 369–370 (1955)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Milman, V.D.: Inégalité de Brunn-Minkowski inverse et applications à la théorie locale des espaces normés. C. R. Acad. Sci. Paris Sér. I Math. 302(1), 25–28 (1986)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Milman, V.D.: Isomorphic symmetrizations and geometric inequalities, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, pp. 107–131 (1988)Google Scholar
  20. 20.
    Milman, V.D.: Geometrization of probability, Geometry and dynamics of groups and spaces, Progr. Math., vol. 265, Birkhäuser, Basel, pp. 647–667 (2008)Google Scholar
  21. 21.
    Milman, V.D., Pajor, A.: Entropy and asymptotic geometry of non-symmetric convex bodies. Adv. Math. 152(2), 314–335 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Naszódi, M.: Fractional illumination of convex bodies. Contrib. Discret. Math. 4(2), 83–88 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Pietsch, A.: Theorie der Operatorenideale (Zusammenfassung), Friedrich-Schiller-Universität, Jena, 1972, Wissenschaftliche Beiträge der Friedrich-Schiller-Universität JenaGoogle Scholar
  24. 24.
    Slomka, B.A.: Covering numbers of log-concave functions and related inequalities, PreprintGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of Mathematics and Computer ScienceThe Open University of IsraelRa’ananaIsrael

Personalised recommendations