A submetric characterization of Rolewicz's property ($β$)

Zhang, Sheng

doi:10.48550/arxiv.2101.08707

Cited by 2 publications

(3 citation statements)

References 14 publications

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“…While writing this article, we learned from Sheng Zhang [Zha21] that he had discovered independently a metric characterization of the class < (β) > in terms of a submetric test-space in the sense of Ostrovskii [Ost14b]. A similar submetric test-space characterization can be extracted with some care from the work of Dilworth, Kutzarova, and Randrianarivony in [DKR16] and is also a direct consequence of our work (see Corollary 3 in Section 2).…”

Section: Introductionmentioning

confidence: 66%

“…Corollary 7 below, which is an immediate consequence of the stability of umbel convexity under nonlinear quotients and Corollary 2, was proved for the first time in [DKR16, Theorem 2.0.1] (for uniform quotients) 3 using the delicate "fork argument" and in [Zha21] (for uniform or coarse quotients) using a more elementary self-improvement argument.…”

Section: Stability Under Nonlinear Quotientsmentioning

confidence: 99%

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Umbel convexity and the geometry of trees

Baudier,

Gartland

2021

Preprint

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For every p ∈ (0, ∞), 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 umbel cotype p, plays a role in obtaining compression rate bounds for coarse embeddings of countably branching trees. Local analogs of these invariants, fork q-convexity and fork cotype q, are introduced and their relationship to Markov q-convexity and relaxations of the q-tripod inequality is discussed. The metric invariants are estimated for a large class of Heisenberg groups. Finally, a new characterization of non-negative curvature is given. Contents 1. Introduction 1 2. Property (β) with power type p implies umbel p-convexity 6 3. Distortion and compression rate of embeddings of countably branching trees 13 4. Stability under nonlinear quotients 17 5. More examples of metric spaces with non-trivial umbel cotype 22 6. Relaxations of the tripod inequality and of Markov convexity 27 7. A characterization of non-negative curvature 39 8. Concluding remarks and open problems 40 References 41

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Section: Introductionmentioning

confidence: 66%

Section: Stability Under Nonlinear Quotientsmentioning

confidence: 99%

Umbel convexity and the geometry of trees

Baudier,

Gartland

2021

Preprint

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“…The doubling subsets are identical to the ones of Bartal-Gottlieb-Neiman and they are described in Section 2.1. The proof uses a self-improvement argument, which was first employed for metric embedding purposes by Johnson and Schechtman in [JS09], and subsequently in [Klo14], [BZ16], [BCD + 17], [Swi18], and [Zha21]; and is carried over in Section 2.2. Our proof has several advantages.…”

Section: Introductionmentioning

confidence: 99%

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

Baudier,

Swieçicki,

Swift

2021

Preprint

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We revisit the main results from [BGN14, BGN15] and [LN14] about the impossibility of dimension reduction for doubling subsets of ℓ q for q > 2. We provide an alternative elementary proof of this impossibility result that combines the simplicity of the construction in [BGN14,BGN15] with the generality of the approach in [LN14] (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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A submetric characterization of Rolewicz's property ($β$)

Abstract: The main result is a submetric characterization of the class of Banach spaces admitting an equivalent norm with Rolewicz's property (β). As applications we prove that up to renorming, property (β) is stable under coarse Lipschitz embeddings and coarse quotients.

Cited by 2 publications

References 14 publications

Umbel convexity and the geometry of trees

Umbel convexity and the geometry of trees

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

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