Weak near convexity and smoothness of Banach spaces

In this paper we study the weak near convexity and smoothness in Banach spaces. These concepts are introduced by using the De Blasi measure of weak noncompactness which is the weak translation of the Hausdorff measure of noncompactness. The De Blasi measure of weak noncompactness fails the isometry invariance property and this fact makes that some results about the near convexity and smoothness in Banach spaces cannot be adapted in the weak version. Particularly, we prove that the weak near smoothness is a property which is transmitted to closed subspaces by using the classical double-limit criterion of Eberlein on the characterization of relatively weakly compact subsets. Moreover, we analyse the relationship between the weak near smoothness in the dual space and the weak near convexity in the original space and, finally, we study some classic Banach spaces in order to illustrate the introduced concepts.