arXiv subjects

Florent P. Baudier

Publications and source records attributed to Florent P. Baudier.

17 recordsLinked to original sources

arXiv2026

The Kadets--Werner modification of Bourgain--Rosenthal space is asymptotically midpoint uniformly convex

Let $X_{\mathsf{KW}}$ be the Kadets--Werner modification of a closed subspace of $L_1$ constructed by Bourgain and Rosenthal in 1980. In this short note, it is shown that $X_{\mathsf{KW}}$ is asymptotically midpoint uniformly convex. Since $X_{\mathsf{KW}}$ has the Daugavet Property, it fails the Point of Continuity Property and thus does not admit an equivalent norm that is asymptotically uniformly convex. Therefore, the previously known properties of $X_{\mathsf{KW}}$ and the new geometric observation answer, in the negative, the question of Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov whether asymptotic midpoint uniform convexity and asymptotic uniform convexity are isomorphically equivalent and a question of Perreau whether asymptotic midpoint uniform convexity implies the Point of Continuity Property.

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arXiv2025

Asymptotic and nonlinear geometries of Banach spaces and their interactions

This book discusses the interactions between the (nonlinear) metric structure of Banach spaces and their linear asymptotic behavior. The overarching problem is to understand how the various linear structures of a Banach space are preserved under certain nonlinear maps. The first chapters contain what are by now classical results to study the most basic and fundamental rigidity problems: the Lipschitz or uniform classification of Banach spaces. The other chapters form the main contribution of this book. The intended goal is to cover the work of many researchers, in particular their discoveries from the past 25 years, trying to understand how asymptotic properties of Banach spaces are preserved under several essential notions of nonlinear (bi-Lipschitz, coarse-Lipschitz, coarse or uniform) embeddings. This is part of a broader program called the Kalton program. This program, inspired by the Ribe program, seeks to uncover purely metric characterizations of asymptotic properties of Banach spaces. Many of these charaterizations are closely connected to the geometry of families of metric graphs (trees, Hamming graphs, diamond graphs, interlacing graphs) thus this book is also about the geometric structure of those graphs.

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arXiv2025

Umbel convexity and the geometry of trees

For every $p\in(0,\infty)$, a new metric invariant called umbel $p$-convexity is introduced. The asymptotic notion of umbel convexity captures the geometry of countably branching trees, much in the same way as Markov convexity, the local invariant which inspired it, captures the geometry of bounded degree trees. Umbel convexity is used to provide a ``Poincaré-type" metric characterization of the class of Banach spaces that admit an equivalent norm with Rolewicz's property $(β)$. We explain how a relaxation of umbel $p$-convexity, called infrasup-umbel $p$-convexity, plays a role in obtaining compression rate bounds for coarse embeddings of countably branching trees. Local analogues of these invariants - fork $p$-convexity and infrasup-fork $p$-convexity - are introduced, and their relationship to Markov $p$-convexity and relaxations of the $p$-fork inequality is discussed. The metric invariants are estimated for a large class of Heisenberg groups, and in particular a parallelogram $p$-convexity inequality is proved for Heisenberg groups over $p$-uniformly convex Banach spaces. Finally, a new characterization of non-negative curvature is given.

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arXiv2024

An asymptotic analog of a local-to-global phenomenon for uniformly convex renormings

In this note, we investigate the renorming theory of Banach spaces with property $(β)$ of Rolewicz. In particular, we give a "coordinate-free" proof of the fact that every Banach space with property $(β)$ admits an equivalent norm that is asymptotically uniformly smooth; a result originally due to Kutzarova for spaces with a Schauder basis. We also show that if a natural modulus associated with a Banach space $X$ with property $(β)$ is positive at some point in the interval $(0,1)$, then $X$ admits an equivalent norm with property $(β)$. This is an asymptotic analog of a profound result from the local geometry of Banach spaces that states that if the modulus of uniform convexity of a Banach space $X$ is positive at some point in the interval $(0,2)$, then $X$ admits an equivalent norm that is uniformly convex.

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arXiv2023

Abstract embeddability ranks

We describe several ordinal indices that are capable of detecting, according to various metric notions of faithfulness, the embeddability between pairs of Polish spaces. These embeddability ranks are of theoretical interest but seem difficult to estimate in practice. Embeddability ranks, which are easier to estimate in practice, are embeddability ranks generated by Schauder bases. These embeddability are inspired by the nonlinear indices à la Bourgain from \cite{BLMS_FM}. In particular, we resolve a problem \cite[Problem 3.10]{BLMS_FM} regarding the necessity of additional set-theoretic axioms regarding the main coarse universality result of \cite{BLMS_FM}.

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arXiv2023

$L_1$-distortion of Wasserstein metrics: a tale of two dimensions

By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid $\{0,1,\dots n\}^2$ has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log n}$. We provide a new "dimensionality" interpretation of Kislyakov's argument, showing that, if $\{G_n\}_{n=1}^\infty$ is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number $δ\in [2,\infty)$, then the 1-Wasserstein metric over $G_n$ has $L_1$-distortion bounded below by a constant multiple of $(\log |G_n|)^{\frac{1}δ}$. We proceed to compute these dimensions for $\oslash$-powers of certain graphs. In particular, we get that the sequence of diamond graphs $\{\mathsf{D}_n\}_{n=1}^\infty$ has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over $\mathsf{D}_n$ has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log| \mathsf{D}_n|}$. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of $L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not $L_1$-embeddable.

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arXiv2023

Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras

We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated to a uniformly locally finite metric space $X$. Under weak assumptions, these $\mathrm{C}^*$-algebras contain embedded copies of $\prod_{k}\mathrm{M}_{n_k}(\mathbb C)$ for any \emph{bounded} countable (possibly finite) collection $(n_k)_k$ of natural numbers; we aim to show that they cannot contain any other von Neumann algebras. One of our main results shows that $L_\infty[0,1]$ does not embed into any of those algebras, even by a not-necessarily-normal $*$-homomorphism. In particular, it follows from the structure theory of von Neumann algebras that any von Neumann algebra which embeds into such algebra must be of the form $\prod_{k}\mathrm{M}_{n_k}(\mathbb C)$ for some countable (possibly finite) collection $(n_k)_k$ of natural numbers. Under additional assumptions, we also show that the sequence $(n_k)_k$ has to be bounded: in other words, the only embedded von Neumann algebras are the ``obvious'' ones.

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arXiv2022

Uniform Roe algebras of uniformly locally finite metric spaces are rigid

We show that if $X$ and $Y$ are uniformly locally finite metric spaces whose uniform Roe algebras, $\cstu(X)$ and $\cstu(Y)$, are isomorphic as \cstar-algebras, then $X$ and $Y$ are coarsely equivalent metric spaces. Moreover, we show that coarse equivalence between $X$ and $Y$ is equivalent to Morita equivalence between $\cstu(X)$ and $\cstu(Y)$. As an application, we obtain that if $Γ$ and $Λ$ are finitely generated groups, then the crossed products $\ell_\infty(Γ)\rtimes_rΓ$ and $ \ell_\infty(Λ)\rtimes_rΛ$ are isomorphic if and only if $Γ$ and $Λ$ are bi-Lipschitz equivalent.

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arXiv2021

No dimension reduction for doubling subsets of $\ell_q$ when $q>2$ revisited

We revisit the main results from \cites{BGN_SoCG14,BGN_SIAM15} and \cite{LafforgueNaor14_GD} about the impossibility of dimension reduction for doubling subsets of $\ell_q$ for $q>2$. We provide an alternative elementary proof of this impossibility result that combines the simplicity of the construction in \cites{BGN_SoCG14,BGN_SIAM15} with the generality of the approach in \cite{LafforgueNaor14_GD} (except for $L_1$ targets). One advantage of this different approach is that it can be naturally generalized to obtain embeddability obstructions into non-positively curved spaces or asymptotically uniformly convex Banach spaces.

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arXiv2020

Stochastic approximation of lamplighter metrics

We observe that embeddings into random metrics can be fruitfully used to study the $L_1$-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the $L_1$-distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on a $n$-point metric space embeds bi-Lipschitzly into $L_1$ with distortion $O(\log n)$. In particular, for every finite group $G$ the lamplighter group $H = \mathbb{Z}_2\wr G$ bi-Lipschitzly embeds into $L_1$ with distortion $O(\log\log|H|)$. In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into $L_1$ of the lamplighter group over the $d$-dimensional infinite lattice $\mathbb{Z}^d$ can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite $d$-dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.

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arXiv2020

On the bi-Lipschitz geometry of lamplighter graphs

In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most~$6$. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into $\ell_1$. We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterizations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it. An application is a characterization in terms of a sequence of graphs with uniformly bounded degree of the notion of trivial Bourgain-Milman-Wolfson type for arbitrary metric spaces, similar to Ostrovskii's characterization previously obtained in \cite{ostrovskii:11}.

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arXiv2020

The geometry of Hamming-type metrics and their embeddings into Banach spaces

Within the class of reflexive Banach spaces, we prove a metric characterization of the class of asymptotic-$c_0$ spaces in terms of a bi-Lipschitz invariant which involves metrics that generalize the Hamming metric on $k$-subsets of $\mathbb{N}$. We apply this characterization to show that the class of separable, reflexive, and asymptotic-$c_0$ Banach spaces is non-Borel co-analytic. Finally, we introduce a relaxation of the asymptotic-$c_0$ property, called the asymptotic-subsequential-$c_0$ property, which is a partial obstruction to the equi-coarse embeddability of the sequence of Hamming graphs. We present examples of spaces that are asymptotic-subsequential-$c_0$. In particular $T^*(T^*)$ is asymptotic-subsequential-$c_0$ where $T^*$ is Tsirelson's original space.

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arXiv2020

Coarse and Lipschitz universality

In this paper we provide several \emph{metric universality} results. We exhibit for certain classes $\cC$ of metric spaces, families of metric spaces $(M_i, d_i)_{i\in I}$ which have the property that a metric space $(X,d_X)$ in $\cC$ is coarsely, resp. Lipschitzly, universal for all spaces in $\cC$ if the collection of spaces $(M_i,d_i)_{i\in I}$ equi-coarsely, respectively equi-Lipschitzly, embeds into $(X,d_X)$. Such families are built as certain Schreier-type metric subsets of $\co$. We deduce a metric analog to Bourgain's theorem, which generalized Szlenk's theorem, and prove that a space which is coarsely universal for all separable reflexive asymptotic-$c_0$ Banach spaces is coarsely universal for all separable metric spaces. One of our coarse universality results is valid under Martin's Axiom and the negation of the Continuum Hypothesis. We discuss the strength of the universality statements that can be obtained without these additional set theoretic assumptions. In the second part of the paper, we study universality properties of Kalton's interlacing graphs. In particular, we prove that every finite metric space embeds almost isometrically in some interlacing graph of large enough diameter.

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arXiv2016

Lipschitz Embeddings of Metric Spaces into $c_0$

Let $M$ be a separable metric space. We say that $f=(f_n):M\to c_0$ is a good-$λ$-embedding if, whenever $x,y\in M$, $x\ne y$ implies $d(x,y)\le\Vert f(x)-f(y)\Vert$ and, for each $n$, $Lip(f_n)<λ$, where $Lip(f_n)$ denotes the Lipschitz constant of $f_n$. We prove that there exists a good-$λ$-embedding from $M$ into $c_0$ if and only if $M$ satisfies an internal property called $π(λ)$. As a consequence, we obtain that for any separable metric space $M$, there exists a good-$2$-embedding from $M$ into $c_0$. These statements slightly extend former results obtained by N. Kalton and G. Lancien, with simplified proofs.

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arXiv2016

On the geometry of the countably branching diamond graphs

In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs $(D_k^ω)_{k\in\mathbb{N}}$ is investigated. In particular it is shown that for every $\varepsilon>0$ and $k\in\mathbb{N}$, $D_k^ω$ embeds bi-Lipschiztly with distortion at most $6(1+\varepsilon)$ into any reflexive Banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm. On the other hand it is shown that the sequence $(D_k^ω)_{k\in\mathbb{N}}$ does not admit an equi-bi-Lipschitz embedding into any Banach space that has an equivalent asymptotically midpoint uniformly convex norm. Combining these two results one obtains a metric characterization in terms of graph preclusion of the class of asymptotically uniformly convexifiable spaces, within the class of separable reflexive Banach spaces with an unconditional asymptotic structure. Applications to bi-Lipschitz embeddability into $L_p$-spaces and to some problems in renorming theory are also discussed.

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arXiv2015

$(β)$-distortion of some infinite graphs

A distortion lower bound of $Ω(\log(h)^{1/p})$ is proven for embedding the complete countably branching hyperbolic tree of height $h$ into a Banach space admitting an equivalent norm satisfying property $(β)$ of Rolewicz with modulus of power type $p\in(1,\infty)$ (in short property ($β_p$)). Also it is shown that a distortion lower bound of $Ω(\ell^{1/p})$ is incurred when embedding the parasol graph with $\ell$ levels into a Banach space with an equivalent norm with property ($β_p$). The tightness of the lower bound for trees is shown adjusting a construction of Matoušek to the case of infinite trees. It is also explained how our work unifies and extends a series of results about the stability under nonlinear quotients of the asymptotic structure of infinite-dimensional Banach spaces. Finally two other applications regarding metric characterizations of asymptotic properties of Banach spaces, and the finite determinacy of bi-Lipschitz embeddability problems are discussed.

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arXiv2013

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 $\ell_q$ into $\ell_p$ ($0 2$. Relevant to geometric group theory purposes, the exact $\ell_p$-compressions of $\ell_2$ are computed. Finally coarse deformation of metric spaces with property A and locally compact amenable groups is investigated.

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