Asymptotic smoothness and concentration properties in Banach spaces.

par Audrey Fovelle (Grenade)

mardi 8 avr. 2025, 14:00 → 15:00 Europe/Paris

Fokko (ICJ)

In 2008, in order to show that $L_p$ is not uniformly homeomorphic to ${\ell }_p\oplus {\ell }_2$ ofr $1<p\ne 2<\mathrm{\infty }$, Kalton and Randrianarivony introduced a new technique based on a certain class of graphs and asymptotic smoothness ideas. More specifically, they proved that reflexive asymptotically uniformly smooth Banach spaces satisfy some concentration property for Lipschitz maps defined on the Hamming graphs. It was proved several years later by Lancien and Raja that quasi-reflexive asymptotically uniformly smooth Banach spaces satisfy a similar concentration property. After introducing all the objects at stake and explaining their interest, we will see how one can construct the first example of a Banach space that has such concentration property without being asymptotically smooth.