Asymptotic smoothness, concentration properties in Banach spaces and applications

We prove an optimal result of stability under ${\mathrm{\ell }}_p$-sums of some concentration properties for Lipschitz maps defined on Hamming graphs into Banach spaces. As an application, we give examples of spaces with Szlenk index arbitrarily high that admit nevertheless a concentration property. In particular, we get the very first examples of Banach spaces with concentration but without asymptotic smoothness property.