Metric entropy of convex hulls in Banach spaces

The paper presents diverse methods for estimating the covering number of a precompact subset of a Banach space when the entropy of the set of its extremal points is already known. In the case of a Hilbert space, the Gelfand diameters of the subset are also estimated. The Krein–Milman theorem is a powerful tool in analysis. The aim of this paper is to quantify this theorem in terms of entropy numbers. More precisely, if we have information about the entropy of a precompact set in a Banach or a Hilbert space, then what can be said about the entropy of its convex hull? We give sharp estimate