Entropy of Absolute Convex Hulls in Hilbert Spaces

The metric entropy of absolute convex hulls of sets in Hilbert spaces is studied for the general case when the metric entropy of the sets is arbitrary. Under some regularity assumptions, the results are sharp. The Krein–Milman theorem is a powerful tool in analysis. To quantify this theorem, a number of researchers have studied the entropy numbers of the convex hulls of precompact sets in a Banach space or a Hilbert space. The goal is to obtain a sharp upper bound for the entropy of the convex hull conv(T ), knowing the entropy of the set T . The importance of this problem was addressed by Dudley in [7], where some special cases were studied. Dudley’s results were improved by Ball and Pajor [2] and Carl [4], and extended to Banach spaces by Carl, Kyrezi and Pajor [5]. Recall that

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