Florent Baudier

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

Abstract.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 uniform 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 the metrics dp induced by the ℓp norms.

A NEW METRIC INVARIANT FOR BANACH SPACES

We show that if the Szlenk index of a Banach space X or of its dual is larger than the first infinite ordinal, 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.

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.

Quantitative nonlinear embeddings into Lebesgue sequence spaces

In this paper fundamental nonlinear geometries of Lebesgue sequence spaces are studied in their quantitative aspects. Applications of this work are a positive solution to the strong embeddability problem from

Embeddability of Snowflaked Metrics with Applications to the Nonlinear Geometry of the Spaces L p and ℓ p for 0<p<∞

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The coarse geometry of Tsirelson’s space and applications

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On the geometry of the countably branching diamond graphs

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(β)-distortion of some infinite graphs

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Tight Embeddability of Proper and Stable Metric Spaces

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Embeddings of proper metric spaces into Banach spaces

. Relevant to geometric group theory purposes, the exact