Nonlinear aspects of super weakly compact sets

We study the notion of super weakly compact subsets of a Banach space, which can be described as a local version of super-reflexivity. Our first result is that the closed convex hull of a super weakly compact set is super weakly compact. This allows us to extend to the non convex setting the main properties of these sets. In particular, we give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.