  These notes concern the nonlinear geometry of Banach spaces, asymptotic
uniform smoothness and several Banach-Saks-like properties. We study the
existence of certain concentration inequalities in asymptotically uniformly
smooth Banach spaces as well as weakly sequentially continuous coarse
(Lipschitz) embeddings into those spaces. Some results concerning the
descriptive set theoretical complexity of those properties are also obtained.
We finish the paper with a list of open problem.
