# Metric entropy of convex hulls in Hilbert spaces

Studia Mathematica (2000)

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

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topLi, 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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- [13] M. Ledoux and M. Talagrand, Probability in a Banach Space, Springer, 1991. Zbl0748.60004
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