Metric entropy of convex hulls in Hilbert spaces

Wenbo Li; Werner Linde

Studia Mathematica (2000)

  • Volume: 139, Issue: 1, page 29-45
  • ISSN: 0039-3223

Abstract

top
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 .

How to cite

top

Li, Wenbo, and Linde, Werner. "Metric entropy of convex hulls in Hilbert spaces." Studia Mathematica 139.1 (2000): 29-45. <http://eudml.org/doc/216709>.

@article{Li2000,
abstract = {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 $(a_j)_\{j=1\}^∞$.},
author = {Li, Wenbo, Linde, Werner},
journal = {Studia Mathematica},
keywords = {metric entropy; convex hull; majorizing measure; Gaussian process; Hilbert space},
language = {eng},
number = {1},
pages = {29-45},
title = {Metric entropy of convex hulls in Hilbert spaces},
url = {http://eudml.org/doc/216709},
volume = {139},
year = {2000},
}

TY - JOUR
AU - Li, Wenbo
AU - Linde, Werner
TI - Metric entropy of convex hulls in Hilbert spaces
JO - Studia Mathematica
PY - 2000
VL - 139
IS - 1
SP - 29
EP - 45
AB - 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 $(a_j)_{j=1}^∞$.
LA - eng
KW - metric entropy; convex hull; majorizing measure; Gaussian process; Hilbert space
UR - http://eudml.org/doc/216709
ER -

References

top
  1. [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. Zbl0746.60005
  2. [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. Zbl1021.60030
  3. [3] B. Carl, Entropy numbers, s-numbers, and eigenvalue problems, J. Funct. Anal. 41 (1981), 290-306. Zbl0466.41008
  4. [4] B. Carl, Metric entropy of convex hulls in Hilbert spaces, Bull. London Math. Soc. 29 (1997), 452-458. Zbl0879.41012
  5. [5] B. Carl and D. E. Edmunds, Entropy of C(X)-valued operators and diverse applications, preprint, 1998. Zbl1031.41013
  6. [6] B. Carl, I. Kyrezi and A. Pajor, Metric entropy of convex hulls in Banach spaces, J. London Math. Soc., to appear. Zbl0976.46009
  7. [7] B. Carl and I. Stephani, Entropy, Compactness and Approximation of Operators, Cambridge Univ. Press, Cambridge, 1990. 
  8. [8] R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Funct. Anal. 1 (1967), 290-330. Zbl0188.20502
  9. [9] R. M. Dudley, Universal Donsker classes and metric entropy, Ann. Probab. 15 (1987), 1306-1326. Zbl0631.60004
  10. [10] X. Fernique, Fonctions aléatoires gaussiennes, vecteurs aléatoires gaussiens, Les Publications CRM, Montréal, 1997. 
  11. [11] A. Garnaev and E. Gluskin, On diameters of the Euclidean ball, Dokl. Akad. Nauk SSSR 277 (1984), 1048-1052 (in Russian). Zbl0588.41022
  12. [12] M. Ledoux, Isoperimetry and Gaussian analysis, in: Lectures on Probability Theory and Statistics, Lecture Notes in Math. 1648, Springer, 1996, 165-294. 
  13. [13] M. Ledoux and M. Talagrand, Probability in a Banach Space, Springer, 1991. Zbl0748.60004
  14. [14] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge, 1989. Zbl0698.46008
  15. [15] S C. Schütt, Entropy numbers of diagonal operators between symmetric Banach spaces, J. Approx. Theory 40 (1984), 121-128. Zbl0497.41017
  16. [16] I. Steinwart, Entropy of C(K)-valued operators, J. Approx. Theory, to appear. Zbl0957.47021
  17. [17] M. Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), 99-149. Zbl0712.60044
  18. [18] M. Talagrand, Majorizing measures: The generic chaining, Ann. Probab. 24 (1996), 1049-1103. Zbl0867.60017

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.