A New Coarsely Rigid Class of Banach Spaces

Baudier, Florent P.; Lancien, Gilles; Motakis, Pavlos; Schlumprecht, Thomas

doi:10.1017/s1474748019000732

Cited by 11 publications

(13 citation statements)

References 34 publications

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“…The next proposition is based on a weak * -compactness argument and will be crucial for our proofs. Although the distance considered on [N] k is different, the proof follows the same lines as Lemma 4.1 in [3]. We therefore state it now without further detail.…”

Section: The Kalton Interlaced Graphs and Propertymentioning

confidence: 92%

On the Coarse Geometry of James Spaces

Lancien¹,

Petitjean²,

Procházka³

2019

Can. Math. Bull.

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In this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space J nor into its dual J * . It is a particular case of a more general result on the non equicoarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic stucture. This allows us to exhibit a coarse invariant for Banach spaces, namely the non equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property Q of Kalton. We conclude with a remark on the coarse geometry of the James tree space J T and of its predual.

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Section: The Kalton Interlaced Graphs and Propertymentioning

confidence: 92%

On the Coarse Geometry of James Spaces

Lancien¹,

Petitjean²,

Procházka³

2019

Can. Math. Bull.

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“…The above idea of [88] (partially building on [155]) inspired a series of investigations [88,91,90,17,104,16] over recent years that led to major coarse non-embeddability results for certain Banach spaces, starting with Kalton's incorporation [88] of Ramsey-theoretic reasoning which led to (among other things) his proof in [88] that c 0 does not embed coarsely into any reflexive Banach space. We will not survey these ideas here, and only state that they rely on the linear theory in multiple ways, so their relevance to the setting of Theorem 1 is questionable.…”

Section: 7mentioning

confidence: 99%

Nonpositive curvature is not coarsely universal

Eskenazis

Mendel

2019

Invent. math.

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We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni, Naor and Neiman (2015). We establish this statement by proving that a metric space which is q-barycentric for some q ∈ [1,∞) has metric cotype q with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the ℓ ∞ grid [m] n ∞ = ({1,... ,m} n , · ∞ ) into ℓ q for all q ∈ (2,∞), from which we deduce the following discrete converse to the fact that ℓ n ∞ embeds with distortion O(1) into ℓ q for q = O(logn). A rigidity theorem of Ribe (1976) implies that for every n ∈ N there exists m ∈ N such that if [m] n ∞ embeds into ℓ q with distortion O(1), then q is necessarily at least a universal constant multiple of log n. Ribe's theorem does not give an explicit upper bound on this m, but by the work of Bourgain (1987) it suffices to take m = n, and this was the previously best-known estimate for m. We show that the above discretization statement actually holds when m is a universal constant.

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“…Given k ∈ N and a directed set I , define A directed set I is said to have infinite tail if Succ (u) is infinite for all u ∈ I . 4 If I has infinite tail, the relation defined above defines a directed partial order on [I ] k . 5 Letā = (a 1 , .…”

Section: Directed Setsmentioning

confidence: 99%

“…It has been recently proven that the class of reflexive asymptotic-ℓ ∞ Banach spaces is stable under coarse embeddings (see [BLMS18], Theorem A). However, the following remains open.…”

Section: Introductionmentioning

confidence: 99%

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

Braga¹

2019

J. Inst. Math. Jussieu

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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 spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.

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A New Coarsely Rigid Class of Banach Spaces

Cited by 11 publications

References 34 publications

On the Coarse Geometry of James Spaces

On the Coarse Geometry of James Spaces

Nonpositive curvature is not coarsely universal

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

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