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

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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 ∞ .

How to cite

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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

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  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
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  18. [18] M. Talagrand, Majorizing measures: The generic chaining, Ann. Probab. 24 (1996), 1049-1103. Zbl0867.60017

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