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 K0⊆X such that, for every weakly compact K⊆X and every ε>0, there is n∈N such that K⊆nK0+εBX. Let μ be a probability measure. It is an open problem whether the Lebesgue-Bochner space L1(μ,X) is SWCG whenever X is SWCG. We prove that L1(μ,X) is SWCG if and only if the family of all uniformly bounded, weakly compact subsets of L1(μ,X) is strongly generated. We show that L1(μ,X) is SWCG if X is a SWCG subspace of an L1 space. For 1