Skip to main content
SpringerLink
Log in

Isomorphic Steiner symmetrization

Inventiones mathematicae volume 153, pages 463–485 (2003)Cite this article

Abstract

This paper proves that there exist 3n Steiner symmetrizations that transform any convex set K⊂ℝn into an isomorphic Euclidean ball; i.e. if vol(K)=vol(D n ) where D n is the standard Euclidean unit ball, then K can be transformed into a body K such that c 1 D n ⊂K⊂c 2 D n , where c 1,c 2 are numerical constants. Moreover, for any c>2, cn symmetrizations are also enough.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Ball, K.M.: An Elementary Introduction to Modern Convex Geometry. Appeared in Flavors of Geometry, S. Levy (ed.), MSRI vol. 31. Cambridge Univ. Press 1997

  2. Blaschke, W.: Über affine Geometrie XIV: Eine Minimumaufgabe für Legendres Trägheitsellipsoid. Ber. Verh. Sächs. Ges. Wiss. Leipzig, Math.-Phys. Kl. 70, 72–75 (1918); Ges. Werke, Band 3, pp. 307–310, Essen: Thales 1985

    Google Scholar 

  3. Brunn, H.: Über Ovale und Eiflächen. Dissertation, München 1887

  4. Brunn, H.: Referat über eine Arbeit: Exacte Grundlagen für eine Theorie der Ovale. Sitzungsber., Bayer. Akad. Wiss., Math.-Phys. Kl. 24, 93–111 (1894)

    Google Scholar 

  5. Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. Berlin: Springer 1934; English transl. Theory of convex bodies, BCS associates 1987

  6. Bourgain, J., Klartag, B., Milman, V.D.: A reduction of the slicing problem to finite volume ratio bodies. C. R. Acad. Sci. Paris, Ser. I 336, 331–334 (2003)

    Google Scholar 

  7. Bobkov, S.G., Nazarov, F.L.: On convex bodies and log-concave probability measures with unconditional bodies. Geometric aspects of functional analysis (2001–2002). Lect. Notes Math., vol. 1807, pp. 53–69. Berlin: Springer 2003

  8. Bourgain, J., Lindenstrauss, J., Milman, V.D.: Estimates related to Steiner symmetrizations. Geometric aspects of functional analysis (1987–88). Lect. Notes Math., vol. 1376, pp. 264–273. Berlin: Springer 1989

  9. Burago, Yu.D., Zalgaller, V.A.: Geometric inequalities. Springer Ser. Sov. Math. Berlin: Springer 1988

  10. Caratheodory, C., Study, E.: Zwei Beweise des Satzes, dass der Kreis unter allen Figuren gleichen Umfanges den grössten Inhalt hat. Math. Ann. 68, 133–140 (1910)

    MATH  Google Scholar 

  11. Garnaev, A.Yu., Gluskin, E.D.: The widths of a Euclidean ball. Dokl. Akad. Nauk SSSR 277, 1048–1052 (1984); English transl. in Sov. Math. Dokl. 30 (1984)

    MathSciNet  MATH  Google Scholar 

  12. Giannopoulos, A.A., Milman, V.D.: Euclidean structures in finite dimensional normed spaces. Handbook of the Geometry of Banach spaces, Vol. 1, pp. 707–779, W.B. Johnson, J. Lindenstrauss (eds.). Elsevier Science 2001

  13. Hadwiger, H.: Einfache Herleitung der isoperimetrischen Ungleichung für abgeschlossene Punktmengen. Math. Ann. 124, 158–160 (1952)

    MathSciNet  MATH  Google Scholar 

  14. Klartag, B.: 5n Minkowski symmetrizations suffice to arrive at an approximate Euclidean ball. Ann. Math. 156, 947–960 (2002)

    Google Scholar 

  15. Kashin, B.S.: Diameters of some finite-dimensional sets and classes of smooth functions. Izv. Akad. Nauk SSSR, Ser. Mat. Tom 41, 334–351 (1977); English transl. in Math. USSR Izv. 11, 317–333 (1977)

    Google Scholar 

  16. Macbeath, A.M.: An extremal property of the hypersphere. Proc. Camb. Philos. Soc. 47, 245–247 (1951)

    MathSciNet  MATH  Google Scholar 

  17. Mani-Levitska, P.: Random Steiner symmetrizations. Stud. Sci. Math. Hung. 21, 373–378 (1986)

    Google Scholar 

  18. Meyer, M., Pajor, A.: On the Blaschke-Santalo inequaqlity. Arch. Math. 55, 82–93 (1990)

    MathSciNet  MATH  Google Scholar 

  19. Meyer, M., Pajor, A.: Sections of the Unit Ball of l p n. J. Funct. Anal. 80, 109–123 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Milman, V.D.: Almost Euclidean qoutient spaces of subspaces of a finite dimensional normed space. Proc. Am. Math. Soc. 94, 445–449 (1985)

    MathSciNet  MATH  Google Scholar 

  21. Milman, V.D., Pajor, A.: Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. Geometric aspects of functional analysis (1987–88). Lect. Notes Math., vol. 1376, pp. 64–104. Berlin: Springer 1989

  22. Milman, V.D., Pajor, A.: Entropy and Asymptotic Geometry of Non-Symmetric Convex Bodies. Adv. Math. 152, 314–335 (2000)

    Google Scholar 

  23. Milman, V.D., Schechtman, G.: Asymptotic theory of finite-dimensional normed spaces. Lect. Notes Math., vol. 1200. Berlin: Springer 1986

  24. Pisier, G.: The volume of convex bodies and Banach space geometry. Camb. Tracts Math., vol. 94. Cambridge Univ. Press 1997

  25. Schneider, R.: Convex bodies: The Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge Univ. Press 1993

  26. Steiner, J.: Einfacher Beweis der isoperimetrischen Hauptsätze. Crelle J. Reine Angew. Math. 18, 281–296 (1838); Ges. Werke 2, G. Reimer, Berlin, 77–91 (1882), reprint bei Chelsea, Bronx, NY (1972)

    Google Scholar 

  27. Vaaler, J.D.: A geometric inequality with applications to linear forms. Pac. J. Math. 83, 543–553 (1979)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    B. Klartag & V.D. Milman

Authors
  1. B. Klartag
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V.D. Milman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Klartag.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Klartag, B., Milman, V. Isomorphic Steiner symmetrization. Invent. math. 153, 463–485 (2003). https://doi.org/10.1007/s00222-003-0290-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-003-0290-y

Keywords

search

Navigation