Skip to main content

Advertisement

SpringerLink
Log in

Functional Covering Numbers

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  4. Artstein-Avidan, S., Milman, V.: A characterization of the support map. Adv. Math. 223(1), 379–391 (2010)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  6. Artstein-Avidan, S., Raz, O.: Weighted covering numbers of convex sets. Adv. Math. 227(1), 730–744 (2011)

    Article  MathSciNet  Google Scholar 

  7. Artstein-Avidan, S., Slomka, B.A.: On weighted covering numbers and the Levi-Hadwiger conjecture. Israel J. Math. 209(1), 125–155 (2015)

    Article  MathSciNet  Google Scholar 

  8. Ball, K.M.: PhD dissertation, Cambridge

  9. Barvinok, A.: A course in convexity, Graduate Studies in Mathematics, vol. 54, American Mathematical Society, Providence, RI (2002)

  10. Bobkov, S., Madiman, M.: Reverse brunn-minkowski and reverse entropy power inequalities for convex measures. J. Funct. Anal. 262(7), 3309–3339 (2012)

    Article  MathSciNet  Google Scholar 

  11. Bourgain, J., Milman, V.: New volume ratio properties for convex symmetric bodies in \({{\bf R}}^n\). Invent. Math. 88(2), 319–340 (1987)

    Article  MathSciNet  Google Scholar 

  12. Fradelizi, M., Meyer, M.: Increasing functions and inverse Santaló inequality for unconditional functions. Positivity 12(3), 407–420 (2008)

    Article  MathSciNet  Google Scholar 

  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. Hadwiger, H.: Ungelöstes Probleme Nr. 20. Elem. Math. 12(6), 121 (1957)

    MathSciNet  Google Scholar 

  15. Klartag, B., Milman, V.D.: Geometry of log-concave functions and measures. Geom. Dedicata 112(1), 169–182 (2005)

    Article  MathSciNet  Google Scholar 

  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)

  17. Levi, F.W.: Überdeckung eines Eibereiches durch Parallelverschiebung seines offenen Kerns. Arch. Math. (Basel) 6, 369–370 (1955)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

  20. Milman, V.D.: Geometrization of probability, Geometry and dynamics of groups and spaces, Progr. Math., vol. 265, Birkhäuser, Basel, pp. 647–667 (2008)

  21. Milman, V.D., Pajor, A.: Entropy and asymptotic geometry of non-symmetric convex bodies. Adv. Math. 152(2), 314–335 (2000)

    Article  MathSciNet  Google Scholar 

  22. Naszódi, M.: Fractional illumination of convex bodies. Contrib. Discret. Math. 4(2), 83–88 (2009)

    MathSciNet  Google Scholar 

  23. Pietsch, A.: Theorie der Operatorenideale (Zusammenfassung), Friedrich-Schiller-Universität, Jena, 1972, Wissenschaftliche Beiträge der Friedrich-Schiller-Universität Jena

  24. Slomka, B.A.: Covering numbers of log-concave functions and related inequalities, Preprint

Download references

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.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Shiri Artstein-Avidan

  2. Department of Mathematics and Computer Science, The Open University of Israel, 4353701, Ra’anana, Israel

    Boaz A. Slomka

Authors
  1. Shiri Artstein-Avidan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boaz A. Slomka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shiri Artstein-Avidan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Artstein-Avidan, S., Slomka, B.A. Functional Covering Numbers. J Geom Anal 31, 1039–1072 (2021). https://doi.org/10.1007/s12220-019-00310-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-019-00310-3

Keywords

Mathematics Subject Classification

Search

Navigation