On the SWCG property in Lebesgue-Bochner spaces

A Banach space X is called strongly weakly compactly generated (SWCG) if the family of all weakly compact subsets of X is strongly generated, i.e. there exists a weakly compact K-0 subset of X such that, for every weakly compact K subset of X and every epsilon > 0, there is n is an element of N such that K subset of nK(0) + epsilon B-X. Let mu be a probability measure. It is an open problem whether the Lebesgue-Bochner space L-1(mu, X) is SWCG whenever X is SWCG. We prove that L-1(mu, X) is SWCG if and only if the family of all uniformly bounded, weakly compact subsets of L-1 (mu, X) is strongly generated. We show that L-1 (mu, X) is SWCG if X is a. SWCG subspace of an L-1 space. For 1 < p < infinity (and non-trivial mu), we prove that the following statements are equivalent: (i) L-p(mu, X) is SWCG; (ii) the family of all sigma'-compact subsets of L-p (mu, X) is strongly generated; (iii) X is reflexive. (C) 2015 Elsevier B.V. All rights reserved.