Skip to main content
SpringerLink
Log in

Covering Convex Bodies by Cylinders and Lattice Points by Flats

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

Abstract

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 1≤k≤d−1.

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. Ball, K.: Flavors of geometry. In: Levy, S. (ed.) An Elementary Introduction to Modern Convex Geometry. Math. Sci. Res. Inst. Publ., vol. 31, pp. 1–58. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  2. Ball, K.: The plank problem for symmetric bodies. Invent. Math. 104, 535–543 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bang, T.: A solution of the “plank problem”. Proc. Am. Math. Soc. 2, 990–993 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  4. Banaszczyk, W.: Inequalities for convex bodies and polar reciprocal lattices in ℝn, II: application of K-convexity. Discrete Comput. Geom. 16(3), 305–311 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Banaszczyk, W., Litvak, A.E., Pajor, A., Szarek, S.J.: The flatness theorem for nonsymmetric convex bodies via the local theory of Banach spaces. Math. Oper. Res. 24(3), 728–750 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bárány, I., Harcos, G., Pach, J., Tardos, G.: Covering lattice points by subspaces. Period. Math. Hung. 43, 93–103 (2001)

    Article  MATH  Google Scholar 

  7. Bezdek, K., Hausel, T.: On the number of lattice hyperplanes which are needed to cover the lattice points of a convex body. In: Böröczky, K., et al. (eds.) Intuitive Geometry, Proceedings of the 3rd International Conference Held in Szeged, Hungary, 1991. Colloq. Math. Soc. János Bolyai, vol. 63, pp. 27–31. North-Holland, Amsterdam (1994)

    Google Scholar 

  8. Chakerian, G.D.: Inequalities for the difference body of a convex body. Proc. Am. Math. Soc. 18, 879–884 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  9. Figiel, T., Tomczak-Jaegermann, N.: Projections onto Hilbertian subspaces of Banach spaces. Isr. J. Math. 33, 155–171 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  10. Grünbaum, B.: Measures of symmetry for convex sets. In: Proc. Sympos. Pure Math., vol. VII, pp. 233–270. American Mathematical Society, Providence (1963)

    Google Scholar 

  11. John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 187–204. Interscience, New York (1948)

    Google Scholar 

  12. Kannan, R., Lovász, L.: Covering minima and lattice-point-free convex bodies. Ann. Math. 128, 577–602 (1988)

    Article  Google Scholar 

  13. Lewis, D.R.: Ellipsoids defined by Banach ideal norms. Mathematika 26, 18–29 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pisier, G.: The Volume of Convex Bodies and Banach Space Geometry. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  15. Pisier, G.: Holomorphic semi-groups and the geometry of Banach spaces. Ann. Math. 115, 375–392 (1982)

    Article  MathSciNet  Google Scholar 

  16. Rogers, C.A., Shephard, G.C.: Convex bodies associated with a given convex body. J. Lond. Math. Soc. 33, 270–281 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rudelson, M.: Distances between non-symmetric convex bodies and the MM *-estimate. Positivity 4, 161–178 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Talata, I.: Covering the lattice points of a convex body with affine subspaces. Bolyai Soc. Math. Stud. 6, 429–440 (1997)

    MathSciNet  Google Scholar 

  19. Tomczak-Jaegermann, N.: Banach-Mazur Distances and Finite-Dimensional Operator Ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38. Longman Scientific & Technical, Harlow (1989). Copublished in the United States with John Wiley & Sons, Inc., New York, 1989

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, 2500 University drive N.W., Calgary, AB, T2N 1N4, Canada

    K. Bezdek

  2. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    A. E. Litvak

Authors
  1. K. Bezdek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. E. Litvak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. E. Litvak.

Additional information

K. Bezdek is partially supported by the Hung. Nat. Sci. Found (OTKA), grant No. T043556.

K. Bezdek and A.E. Litvak are partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bezdek, K., Litvak, A.E. Covering Convex Bodies by Cylinders and Lattice Points by Flats. J Geom Anal 19, 233–243 (2009). https://doi.org/10.1007/s12220-008-9063-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-008-9063-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation