Metrical characterization of super-reflexivity and linear type of Banach spaces

We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain's result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniforms embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with a suitable metric.Comment: to appear in Archiv der Mathemati

Internet Sales Taxes From Borders to Amazon: How Long Before All of Your Purchases Are Taxed?

What so many internet consumers believe to be tax-free is actually subject to a state use tax. Faced with pressure from states that realize very little of the use tax owed, many online retailers, such as Wal-mart, voluntarily collect sales taxes from their customers. But a recent California Appeals Court decision, Borders Online v. State Board of Equalization, could mark a shift towards more prevalent, if not universal, taxation of internet retail

Embeddings of locally finite metric spaces into Banach spaces

We show that if X is a Banach space without cotype, then every locally finite metric space embeds metrically into X.Comment: 6 pages, to appear in Proceedings of the AM

Tight embeddability of proper and stable metric spaces

We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for $p\in [1,\infty]$, every proper subset of $L_p$ is almost Lipschitzly embeddable into a Banach space $X$ if and only if $X$ contains uniformly the $\ell_p^n$'s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.Comment: 19 page

A new metric invariant for Banach spaces

We show that if the Szlenk index of a Banach space $X$ is larger than the first infinite ordinal $\omega$ or if the Szlenk index of its dual is larger than $\omega$, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into $X$. We show that the converse is true when $X$ is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms.Comment: 22 page