Strongly non-embeddable metric spaces

Kelleher, Casey Lynn; Miller, Daniel; Osborn, Trenton; Weston, Anthony

doi:10.1016/j.topol.2011.11.041

Cited by 4 publications

(3 citation statements)

References 7 publications

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“…They showed that this is so for coarse embeddings into any metric space that satisfies a classical condition, called generalized roundness p ∈ (0, ∞), which was introduced and used by Enflo [44] to answer an old question of Smirnov [65] whether every metric space embeds uniformly into a Hilbert space (strictly speaking, [42] considered only the case p = 2, but the argument works mutatis mutandis for any p > 0). A unified treatment of these approaches, so as to rule out coarse and uniform embeddings simultaneously, appears in [93]. As noted by several authors [158,103,153,93], a "vanilla" application of the approach of [42] fails to apply to targets that are general Alexandrov spaces of nonpositive curvature, i.e., to prove Theorem 1.…”

Section: 7mentioning

confidence: 99%

“…A unified treatment of these approaches, so as to rule out coarse and uniform embeddings simultaneously, appears in [93]. As noted by several authors [158,103,153,93], a "vanilla" application of the approach of [42] fails to apply to targets that are general Alexandrov spaces of nonpositive curvature, i.e., to prove Theorem 1. Indeed, a combination of the characterization of generalized roundness in [109] with the work [51] shows that there are Alexandrov spaces of nonpositive curvature that do not have positive generalized roundness; specifically, this is so for the quaternionic hyperbolic space (and not so for the real and complex hyperbolic spaces [58,51]).…”

Section: 7mentioning

confidence: 99%

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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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Section: 7mentioning

confidence: 99%

Section: 7mentioning

confidence: 99%

Nonpositive curvature is not coarsely universal

Eskenazis

Mendel

2019

Invent. math.

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“…Dranishnikov et al [7] modified Enflo's example to construct a locally finite metric space that is not coarsely embeddable in any Hilbert space, thereby settling a prominent question of Gromov. Kelleher et al [19] unified these examples to construct a locally finite metric space that is not uniformly or coarsely embeddable in any metric space of positive generalised roundness. One may also use generalised roundness as a highly effective isometric invariant by exploiting the connection between generalised roundness and negative type due to Lennard et al [25].…”

Section: Introductionmentioning

confidence: 99%

Comparing the Generalised Roundness of Metric Spaces

Levy-Moore

Nichols

Weston

2016

Bull. Aust. Math. Soc.

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Motivated by the local theory of Banach spaces we introduce a notion of finite representability for metric spaces. This allows us to develop a new technique for comparing the generalized roundness of metric spaces. We illustrate this technique in two different ways by applying it to Banach spaces and metric trees. In the realm of Banach spaces we obtain results such as the following: (1) if U is any ultrafilter and X is any Banach space, then the second dual X * * and the ultrapower (X) U have the same generalized roundness as X, and (2) no Banach space of positive generalized roundness is uniformly homeomorphic to c 0 or ℓp, 2 < p < ∞. Our technique also leads to the identification of new classes of metric trees of generalized roundness one. In particular, we give the first examples of metric trees of generalized roundness one that have finite diameter. These results on metric trees provide a natural sequel to a paper of Caffarelli et al. [6]. In addition, we show that metric trees of generalized roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees.2010 Mathematics Subject Classification. 46B07, 46B80, 05C05, 05C12.

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Metric trees of generalized roundness one

Doust¹,

Weston²

2011

Aequat. Math.

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Abstract. Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric trees that have generalized roundness precisely one.At the outset we consider spherically symmetric trees endowed with the usual path metric (SSTs). Using a simple geometric argument we show how to determine reasonable upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits, it follows that if the downward degree sequencehas generalized roundness one. In particular, all complete n-ary trees of depth ∞ (n ≥ 2), all k-regular trees (k ≥ 3) and all inductive limits of Cantor trees are seen to have generalized roundness one.The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs. It remains an intriguing problem to completely classify countable metric trees of generalized roundness one.

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Strongly non-embeddable metric spaces

Cited by 4 publications

References 7 publications

Nonpositive curvature is not coarsely universal

Nonpositive curvature is not coarsely universal

Comparing the Generalised Roundness of Metric Spaces

Metric trees of generalized roundness one

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