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

In this paper we consider the Wośko 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.