Isometric embeddings of snowflakes into finite-dimensional Banach spaces

Donne, Enrico Le; Rajala, Tapio; Walsberg, Erik

doi:10.1090/proc/13778

Cited by 3 publications

(2 citation statements)

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“…Remarkable results in this direction were obtained by Schoenberg, independently in [Sch38a,Sch38b] and, together with von Neumann in [vNS41], where they determined all metric transforms for which a transformed Euclidean space isometrically embeds into another Euclidean space. See [DL10, Chapter 9]) for a discussion and [LDRW18] for some recent developments.…”

Section: Introductionmentioning

confidence: 99%

Metric transforms yielding Gromov hyperbolic spaces

Dragomir

Nicas

2018

Geom Dedicata

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A real valued function ϕ of one variable is called a metric transform if for every metric space (X, d) the composition dϕ = ϕ • d is also a metric on X. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms ϕ such that the transformed Euclidean half line ([0, ∞), | · |ϕ) is Gromov hyperbolic. As a consequence, we obtain metric transform rigidity for roughly geodesic Gromov hyperbolic spaces, that is, if (X, d) is any metric space containing a rough geodesic ray and ϕ is an approximately nondecreasing, unbounded metric transform such that the transformed space (X, dϕ) is Gromov hyperbolic and roughly geodesic then ϕ is an approximate dilation and the original space (X, d) is Gromov hyperbolic and roughly geodesic.

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Section: Introductionmentioning

confidence: 99%

Metric transforms yielding Gromov hyperbolic spaces

Dragomir

Nicas

2018

Geom Dedicata

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“…Theorem 2 is inspired by the following proposition. Which is a key ingredient for showing that finite dimensional normed spaces do not contain infinite snowflakes, see [8].…”

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confidence: 99%

Sets with small angles in self-contracted curves

Zolotov

2018

Preprint

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We study metric spaces with bounded rough angles. E. Le Donne, T. Rajala and E. Walsberg implicitly used this notion to show that infinite snowflakes can not be isometrically embedded into finite dimensional Banach spaces.We show that bounded non-rectifiable self-contracted curves contain metric subspaces with bounded rough angles. Which provides rectifiability of bounded self-contracted curves in a wide class of metric spaces including reversible C ∞ -Finsler manifolds, locally compact CAT(k)-spaces with locally extendable geodesics and locally compact Busemann spaces with locally extendable geodesics.We also extend the result on non embeddability of infinite snowflakes to this class of spaces.2010 Mathematics Subject Classification. 51F99.

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Self-Contracted Curves in Spaces With Weak Lower Curvature Bound

Lebedeva

Ohta

Zolotov

2020

International Mathematics Research Notices

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We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author’s previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. Here we apply this strategy to spaces with lower curvature bound.

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Isometric embeddings of snowflakes into finite-dimensional Banach spaces

Cited by 3 publications

References 20 publications

Metric transforms yielding Gromov hyperbolic spaces

Metric transforms yielding Gromov hyperbolic spaces

Sets with small angles in self-contracted curves

Self-Contracted Curves in Spaces With Weak Lower Curvature Bound

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