Geometriae Dedicata

, Volume 171, Issue 1, pp 293–301 | Cite as

Isomorphic Steiner symmetrization of \(p\)-convex sets

  • Alexander Segal
Original Paper


In this paper we show that given a \(p\)-convex set \(K \subset \mathbb{R }^n\), there exist \(5n\) Steiner symmetrizations that transform it into an isomorphic Euclidean ball. That is, if \(|K| = |D_n| = \kappa _n\), we may symmetrize it, using \(5n\) Steiner symmetrizations, into a set \(K'\) such that \(c_p D_n \subset K' \subset C_p D_n\), where \(c_p\) and \(C_p\) are constants dependent on \(p\) only.


Symmetrization \(p\)-convexity Approximation by convex sets 

Mathematics Subject Classification

28A05 52A20 52A27 



The author would like to thank Prof. Vitali Milman for suggesting to extend the results in [13] to the \(p\)-convex setting, and for his useful advice and discussions. The author would also like to thank Liran Rotem and the anonymous referee for their useful comments throughout the paper.


  1. 1.
    Bourgain, J., Lindenstrauss, J., Milman, V.D.: Estimates related to Steiner symmetrizations. Geometric aspects of functional analysis (1987–88). In: Lecture Notes in Mathematics, vol. 1376, pp. 264–273. Springer, Berlin (1989)Google Scholar
  2. 2.
    Burchard, A., Fortier, M.: Random polarizations. Adv. Math. (accepted)Google Scholar
  3. 3.
    Caratheodory, C., Study, E.: Zwei Beweise des Satzes, da der Kreis unter allen Figuren gleichen Umfanges den gr oten Inhalt hat. Math. Ann. 68, 133–140 (1909)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Dilworth, S.J.: The dimension of Euclidean subspaces of quasi-normed spaces. Math. Proc. Camb. Philos. Soc. 97, 311 (1975)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Edler, F.: Vervollstndigung der Steinerschen elementar-geometrichen Beweise fr den Satz, da der Kreis greren Flcheninhalt besitzt, als jede andere ebene Figur gleich groen Umfanges. Nachr. Knigl. Ges. Wiss. Gttingen. Math-phys. Kl. pp. 73–80 (1882)Google Scholar
  6. 6.
    Figiel, T., Lindenstrauss, J., Milman, V.D.: The dimension of almost spherical sections of convex bodies. Acta Math. 139, 53–94 (1977)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Gordon, Y., Kalton, N.J.: Local structure theorey for quasi-normed spaces. Bull. Sci. Math. 118, 441–453 (1994)MATHMathSciNetGoogle Scholar
  8. 8.
    Giannopoulos, A.A., Milman, V.D.: Euclidean structure in Finite dimensional normed spaces, survey. In: Handbook of the geometry of banach spaces, vol. 1, PP. 707–779 (2001)Google Scholar
  9. 9.
    Gross, W.: Die Minimaleigenschaft der Kugel. Monatsh. Math. Phys. 28(1), 77–97 (1917)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Gruber, P.: Convex and Discrete Geometry. Springer, New York (2007)MATHGoogle Scholar
  11. 11.
    Hadwiger, H.: Einfache Herleitung der isoperimetrischen Ungleichung fur abgeschlossene Punktmengen. Math. Ann. 124, 158–160 (1952)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Klartag, B.: 5n Minkowski symmetrizations suffice to arrive at an approximate Euclidean ball. Ann. Math. 156(3), 947–960 (2002)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Klartag, B., Milman, V.D.: Isomorphic Steiner symmetrizations. Invent. Math. 153(3), 463–485 (2003)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Mani-Levitska, P.: Random Steiner symmetrizations. Studia Sci. Math. Hungar. 21(3–4), 373–378 (1986)MATHMathSciNetGoogle Scholar
  15. 15.
    Milman, V.D.: New proof of theorem of Dvoretzky on sections of convex bodies. Func. Anal. Appl. 5, 28–37 (1971)MathSciNetGoogle Scholar
  16. 16.
    Milman, V.D.: Isomorphic euclidean regularization of quasi-norms in \({\mathbb{R}}^n\). C.R. Acad. Sci. Paris Serie I 321, 879–884 (1995)MATHMathSciNetGoogle Scholar
  17. 17.
    Schneider, R.: Convex Bodies: The Brunn Minkowski Theorey. Cambridge University Press, Cambridge (1993)Google Scholar
  18. 18.
    Van Schaftingen, J.: Approximation of symmetrizations and symmetry of critical points. Topol. Methods Nonlinear Anal. 28(1), 61–85 (2006)MATHMathSciNetGoogle Scholar
  19. 19.
    Volcic, A.: Random Steiner symmetrizations of sets and functions. Calc. Var. Partial Differ. Equ. 46 (3–4), 555–569 (2012)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Tel Aviv UniversityTel AvivIsrael

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