, Volume 8, Issue 4, pp 401–406 | Cite as

On the Dual Form of ‘Low M* Estimate’ in the Quasi-convex Case

  • A. E. Litvak


Let \(B_{2}^{n}\) denote the Euclidean ball in \({\mathbb R}^n\), and, given closed star-shaped body \(K \subset {\mathbb R}^{n}, M_{K}\) denote the average of the gauge of K on the Euclidean sphere. Let \(p \in (0,1)\) and let \(K \subset {\mathbb R}^{n}\) be a p-convex body. In [17] we proved that for every \(\lambda \in (0,1)\) there exists an orthogonal projection P of rank \((1 - \lambda)n\) such that
$$\frac{f(\lambda)}{M_K} PB^{n}_{2} \subset PK,$$
where \(f(\lambda)=c_p\lambda^{1+1/p}\) for some positive constant c p depending on p only. In this note we prove that \(f(\lambda)\) can be taken equal to \(C_p\lambda^{1/p-1/2}\). In terms of Kolmogorov numbers it means that for every \(k \leq n\)
$$d_k (\hbox{Id}:\ell^{n}_{2} \to ({\mathbb R}^{n},\|\cdot\|_{K})) \leq C_p \frac{n^{1/p-1}}{k^{1/p-1/2}} \ell (\hbox{ID}: \ell^{n}_{2} \to ({\mathbb R}^{n}, \|\cdot\|_{K})),$$
where \(\ell(\hbox{Id})={\bf E}\|\sum\limits^{n}_{i=1}g_i e_i\|_K\) for the independent standard Gaussian random variables \(\{g_i\}\) and the canonical basis \(\{e_i\}\) of \({\mathbb R}^n\). All results do not require the symmetry of K.


\(\ell\)-functional Kolmogorov numbers ‘low M*-estimate’ p-norms Quasi-convexity ‘random’ projections 

Mathematics Subject Classifications (1999). Primary:

46B07 52A30 46A16 


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  1. 1.
    Bastero, J., Bernues, J., Pena, A. 1995An extension ofMilman’s reverse Brunn.Minkowski inequalityGeom. Funct. Anal.5572581Google Scholar
  2. 2.
    Bastero, J., Bernues, J., Pena, A. 1995The theorems ofCaratheodory and Gluskin for 0<p<1Proc. Amer. Math.123141144Google Scholar
  3. 3.
    Dilworth, S.J. 1985The dimension ofEuclidean subspaces ofquasi-normed spacesMath. Proc. Camb. Phil. Soc.97311320Google Scholar
  4. 4.
    Gordon, Y.: On Milman’s inequality and random subspaces which escape through a mesh in \(\mathbb{R}^{n}\) . In: Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., Vol. 1317, Springer, Berlin, New York, 1988, pp. 84.106Google Scholar
  5. 5.
    Gordon, Y., Lewis, D.R. 1991Dvoretzky’s theorem for quasi-normed spacesIllinois J. Math.35250259Google Scholar
  6. 6.
    Gordon, Y., Kalton, N.J. 1994Local structure theory for quasi-normed spacesBull. Sci. Math.118441453Google Scholar
  7. 7.
    Guedon, O. 1999Kahane-Khinchine type inequalities for negative exponentMathematika46165173Google Scholar
  8. 8.
    Guedon, O., Litvak, A.E. Euclidean projections of p-convex body, GAFA, Lecture Notes in Math., Vol. 1745, Springer-Verlag, 2000, 95.108Google Scholar
  9. 9.
    Kalton, N.J. 1981Convexity, type and the three space problemStudia Math.69247287Google Scholar
  10. 10.
    Kalton, N.J. 1986Banach envelopes ofnon-locally convex spaces CanJ. Math.386586Google Scholar
  11. 11.
    Kalton, N.J., Litvak, A.E.: Quotients of.nite-dimensional quasi-normed spaces, Houston J. Math. (to appear)Google Scholar
  12. 12.
    Kalton, N.J., Peck, N.T., Roberts, J.W.: An F-space sampler, London Mathematical Society Lecture Note Series, Vol. 89, Cambridge University Press, Cambridge, New York, 1984Google Scholar
  13. 13.
    Kalton, N.J., Sik-Chung, Tam. 1993Factorization theorems for quasi-normed spacesHouston J. Math.19301317Google Scholar
  14. 14.
    Konig, H.: Eigenvalue distribution of compact operators. Operator theory: advances and applications, Vol. 16, Birkhauser Verlag, Basel, Boston, Mass., 1986Google Scholar
  15. 15.
    Latala R. On the equivalence between geometric and arithmetic means for logconcave measures. In: Convex geometric analysis (Berkeley, CA, 1996), Math. Sci. Res. Inst. Publ., Vol. 34, Cambridge Univ. Press, Cambridge, 1999, pp. 123.127Google Scholar
  16. 16.
    Litvak, A.E. 2000Kahane-Khinchin’s inequality for the quasi-normsCanad. Math. Bull.43368379Google Scholar
  17. 17.
    Litvak, A.E., Milman, V.D., Pajor, A. 1999The covering numbers and M*-estimate’ for quasi-convex bodiesProc. Amer. Math. Soc.12714991507Google Scholar
  18. 18.
    Litvak, A.E., Milman, V.D., Schechtman, G. 1998Averages of norms and quasi-normsMath. Ann.31295124Google Scholar
  19. 19.
    Milman, V.D. 1985Almost Euclidean quotient spaces ofsubspaces ofa .nite dimensional normed spaceProc. Amer. Math. Soc.94445449Google Scholar
  20. 20.
    Milman, V.D.: Random subspaces ofproportional dimension of.nite dimensional normed spaces: approach through the isoperimetric inequality. In: Banach spaces (Columbia, Mo., 1984), Lecture Notes in Math., Vol. 1166, Springer, Berlin, New York, 1985, pp. 106.115Google Scholar
  21. 21.
    Milman, V.D.: A note on a low M*-estimate. In: Geometry of Banach spaces (Strobl, 1989), London Math. Soc. Lecture Note Ser., Vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 219–229Google Scholar
  22. 22.
    Milman, V.D. 1996Isomorphic Euclidean regularization ofquasi-norms in \(\mathbb{R}^{n}\)C. R. Acad. Sci. Paris321879884Google Scholar
  23. 23.
    Milman, V.D. 1985Almost Euclidean quotient spaces ofsubspaces ofa .nite-dimensional normed spaceProc. Amer. Math. Soc.94445449Google Scholar
  24. 24.
    Milman, V.D., Schechtman, G.: Asymptotic theory of .nite-dimensional normed spaces, Lecture Notes in Math., Vol. 1200, Springer, Berlin, New York, 1985Google Scholar
  25. 25.
    Pajor, A., Tomczak-Jaegermann, N. 1986Subspaces ofsmall codimension of.nite-dimensional Banach spacesProc. Amer. Math. Soc.97637642Google Scholar
  26. 26.
    Peck, N.T. 1981Banach-Mazur distances and projections on p-convex spacesMath. Z.177131142Google Scholar
  27. 27.
    Pisier, G. 1989The volume of convex bodies and Banach space geometryCambridge University PressCambridgeGoogle Scholar
  28. 28.
    Rolewicz, S.: Metric linear spaces. Monografie Matematyczne, Tom. 56 [Mathematical Monographs, Vol. 56], PWN-Polish Scientific Publishers, Warsaw, 1972Google Scholar

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© Kluwer Academic Publishers 2005

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmonton, ABCanada

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