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Metric entropy of convex hulls in Hilbert spaces

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Let T be a precompact subset of a Hilbert space. We estimate the metric entropy of co(T), the convex hull of T, by quantities originating in the theory of majorizing measures. In a similar way, estimates of the Gelfand width are provided. As an application we get upper bounds for the entropy of co(T), $T={t_1,t_2,...}$, $||t_j||≤a_j$, by functions of the $a_j$'s only. This partially answers a question raised by K. Ball and A. Pajor (cf. [1]). Our estimates turn out to be optimal in the case of slowly decreasing sequences $(a_j)_{j=1}^∞$.
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autor
  • Department of Mathematics, University of Delaware Newark, DE 19711, U.S.A. , wli@math.udel.edu
autor
Bibliografia
  • [1] K. Ball and A. Pajor, The entropy of convex bodies with "few" extreme points, in: London Math. Soc. Lecture Note Ser. 158, Cambridge Univ. Press, 1990, 25-32.
  • [2] B. Bühler, W. V. Li and W. Linde, Localization of majorizing measures, in: Asymptotic Methods in Probability and Statistics with Applications, Birkhäuser, to appear.
  • [3] B. Carl, Entropy numbers, s-numbers, and eigenvalue problems, J. Funct. Anal. 41 (1981), 290-306.
  • [4] B. Carl, Metric entropy of convex hulls in Hilbert spaces, Bull. London Math. Soc. 29 (1997), 452-458.
  • [5] B. Carl and D. E. Edmunds, Entropy of C(X)-valued operators and diverse applications, preprint, 1998.
  • [6] B. Carl, I. Kyrezi and A. Pajor, Metric entropy of convex hulls in Banach spaces, J. London Math. Soc., to appear.
  • [7] B. Carl and I. Stephani, Entropy, Compactness and Approximation of Operators, Cambridge Univ. Press, Cambridge, 1990.
  • [8] R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Funct. Anal. 1 (1967), 290-330.
  • [9] R. M. Dudley, Universal Donsker classes and metric entropy, Ann. Probab. 15 (1987), 1306-1326.
  • [10] X. Fernique, Fonctions aléatoires gaussiennes, vecteurs aléatoires gaussiens, Les Publications CRM, Montréal, 1997.
  • [11] A. Garnaev and E. Gluskin, On diameters of the Euclidean ball, Dokl. Akad. Nauk SSSR 277 (1984), 1048-1052 (in Russian).
  • [12] M. Ledoux, Isoperimetry and Gaussian analysis, in: Lectures on Probability Theory and Statistics, Lecture Notes in Math. 1648, Springer, 1996, 165-294.
  • [13] M. Ledoux and M. Talagrand, Probability in a Banach Space, Springer, 1991.
  • [14] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge, 1989.
  • [15] S C. Schütt, Entropy numbers of diagonal operators between symmetric Banach spaces, J. Approx. Theory 40 (1984), 121-128.
  • [16] I. Steinwart, Entropy of C(K)-valued operators, J. Approx. Theory, to appear.
  • [17] M. Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), 99-149.
  • [18] M. Talagrand, Majorizing measures: The generic chaining, Ann. Probab. 24 (1996), 1049-1103.
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bwmeta1.element.bwnjournal-article-smv139i1p29bwm
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