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Metric entropy of convex hulls in Banach spaces. (English) Zbl 0976.46009

The \(n\)th entropy number \(\varepsilon_n(A)\) of a bounded set \(A\) is the infimum of those \(\varepsilon\) for which \(A\) can be covered by \(n\) balls of radius \(\varepsilon\). If \(A\) is relatively compact, this sequence obviously converges to zero. The authors’ main purpose is to show that if the sequence of entropy numbers (is dominated by a sequence which) converges to zero reasonably slowly, then the entropy numbers of the convex hull \(\text{ co}(A)\) can be estimated in terms of those of \(A\). The techniques, which involve studying the entropy numbers of linear operators, yield asymptotically sharp results. Stronger estimates are obtained in Banach spaces with non-trivial weak type. For subsets of Hilbert space, estimates for the so-called Gelfand diameters are also given. The paper contains a few misprints. For example, \(s_n(A)\) in Proposition 4.5 should be \(s_n\) and \(\ln N(A;\varepsilon)\) in Proposition 6.1 should be \(\ln N(\text{co}(A);\varepsilon)\).

MSC:

46B99 Normed linear spaces and Banach spaces; Banach lattices
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
46B28 Spaces of operators; tensor products; approximation properties
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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