6533b7cffe1ef96bd125999e

RESEARCH PRODUCT

Proper 1-ball contractive retractions in Banach spaces of measurable functions

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In this paper we consider the Wosko problem of evaluating, in an infinite-dimensional Banach space X, the infimum of all k > 1 for which there exists a k-ball contractive retraction of the unit ball onto its boundary. We prove that in some classical Banach spaces the best possible value 1 is attained. Moreover we give estimates of the lower H-measure of noncompactness of the retractions we construct. 1. Introduction Let X be an infinite-dimensional Banach space with unit closed ball B(X) and unit sphere S(X). It is well known that, in this setting, there is a retraction of B(X) onto S(X), that is, a continuous mapping R : B(X) ! S(X) with Rx = x for all x 2 S(X). In (4) Benyamini and Sternfeld, following Nowak ((13)), proved that such a retraction can be chosen among Lipschitz mappings. The problem of evaluating the infimum k0(X) of the Lipschitz constants of such retractions is of considerable interest in the literature. A general result states that in any Banach space X, 3 6 k0(X) 6 k0 (see (8, 10)), where k0 is a universal constant. In special spaces more precise estimates have been obtained by means of constructions which depend on each space. We refer the reader to (9, 10) for a collection of results on this problem and related ones.