Asymptotic smoothness, concentration properties in Banach spaces and applications Fovelle, Audrey Banach spaces Hamming graphs Asymptotic smoothness Nonlinear embeddings Concentration properties Research partially supported by MCIN/AEI/10.13039/501100011033 grant PID2021-122126NBC31 and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by MCIN/AEI/10.13039/501100011033. We prove an optimal result of stability under ℓp-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. 2025-02-17T09:54:25Z 2025-02-17T09:54:25Z 2024-11-20 journal article Published version: Fovelle, Audrey. Journal of Functional Analysis Volume 288, Issue 4, 15 February 2025, 110763. https://doi.org/10.1016/j.jfa.2024.110763 https://hdl.handle.net/10481/102399 10.1016/j.jfa.2024.110763 eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional Elsevier