Entropy of absolutely convex hulls

Rudolph, Philipp GND

Let us consider a precompact subset A of a Banach space X. Then it is common knowledge that also the real absolutely convex hull aco(A) of A is precompact. The aim of this dissertation is to investigate how the precompactness of A changes when passing from A to the real absolutely convex hull aco(A). For this purpose we need a measure of precompactness. This place is taken by the entropy numbers of a set; the rate of decay of the entropy numbers of a set can be considered as a measure of precompactness of the set. We ask, how the rate of decay of the entropy numbers A affects the rate of decay of the dyadic entropy numbers of aco(A). This problem was first treated in a general form by Dudley in 1987; his research was motivated by applications in the field of empirical processes. In recent years, the entropy of absolutely convex hulls has been intensively studied in different settings. Results essentially depend on the underlying Banach space X and the degree of precompactness of A, expressed by the rate of decay of the entropy number sequence of A. The problem of estimating the entropy of the absolutely convex hull aco(A) has already been examined thoroughly in the Hilbert space case. This work mainly deals with the case that X is a Banach space of type p. Some authors also studied the setting where X is an arbitrary Banach space. As far as the decay of the entropy numbers of A is concerned, we are interested in polynomial decay and logarithmic decay. In addition, we consider the case where the (dyadic) entropy numbers of A belong to some Lorentz sequence space. Due to technical reasons, we will sometimes switch from entropy to covering numbers and vice versa. However, we must also take account of the structure of the set A. This question leads to the so-called diagonal case. This work mainly deals with the non-diagonal case where A is any precompact subset of X and no further information about the structure of A are given.

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Rudolph, Philipp: Entropy of absolutely convex hulls. 2014.

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