Related papers: Asymptotic smoothness, concentration properties in…

In this note, we study some concentration properties for Lipschitz maps defined on Hamming graphs, as well as their stability under sums of Banach spaces. As an application, we extend a result of Causey on the coarse Lipschitz structure of…

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…

We study four asymptotic smoothness properties of Banach spaces, denoted $\textsf{T}_p,\textsf{A}_p, \textsf{N}_p$ and $\textsf{P}_p$. We complete their description by proving the missing renorming theorem for $\textsf{A}_p$. We prove that…

In this paper, we study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotically uniformly smoothness and convexity, and several distinct Banach-Saks-like properties. Among…

We introduce the notion of asymptotic coarse Lipschitz equivalence of metric spaces. We show that it is strictly weaker than coarse Lipschitz equivalence. We study its impact on the asymptotic dimension of metric spaces. Then we focus on…

We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically…

The goal of this note is to prove that every real-valued Lipschitz function on a Banach space can be pointwise approximated on a given $\sigma$-compact set by smooth cylindrical functions whose asymptotic Lipschitz constants are controlled.…

In this paper structure of infinite dimensional Banach spaces is studied by using an asymptotic approach based on stabilization at infinity of finite dimensional subspaces which appear everywhere far away. This leads to notions of…

The new class of Banach spaces, so-called asymptotic $l_p$ spaces, is introduced and it is shown that every Banach space with bounded distortions contains a subspace from this class. The proof is based on an investigation of certain…

In this note, we extend to the setting of quasi-reflexive spaces a classical result of N. Kalton and L. Randrianarivony on the coarse Lipschitz structure of reflexive and asymptotically uniformly smooth Banach spaces. As an application, we…

The purpose of this article is to generalize some known characterizations of Banach space properties in terms of graph preclusion. In particular, it is shown that superreflexivity can be characterized by the non-equi-bi-Lipschitz…

We prove that if $Y$ is a closed subspace of a Banach space $X$ such that $Y$ and $X/Y$ admit an equivalent asymptotically uniformly smooth norm, then $X$ also admits an equivalent asymptotically uniformly smooth norm. The proof is based on…

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it…

We provide a short characterization of $p$-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties…

We prove that the class of reflexive asymptotic-$c_0$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic-$c_0$ space $Y$, then $X$ is also reflexive and asymptotic-$c_0$. In order…

We study the asymptotics of strongly continuous operator semigroups defined on locally convex spaces in order to develop a stability theory for solutions of evolution equations beyond Banach spaces. In the classical case, there is only…

We give a quantitative result about asymptotic moduli of Banach spaces under coarse quotient maps. More precisely, we prove that if a Banach space $Y$ is a coarse quotient of a subset of a Banach space $X$, where the coarse quotient map is…

We study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spaces $\ell_p^d$: they behave as $\sqrt[p]{d}$ if $1\le p\le 2$ and as $\sqrt{d}$ if…

We study Banach spaces X with a strongly asymptotic l_p basis (any disjointly supported finite set of vectors far enough out with respect to the basis behaves like l_p) which are minimal (X embeds into every infinite dimensional subspace).…

For $1<p\leqslant \infty$, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $\textsf{A}_p$ and $\textsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show…