scholarly journals Bilipschitz embeddings of metric spaces into Euclidean spaces

1999 ◽  
Vol 43 ◽  
pp. 571-653 ◽  
Author(s):  
S. Semmes
Keyword(s):  
Metric Spaces ◽  
Euclidean Spaces

scholarly journals From Euclidean Spaces to Metric Spaces

2019 ◽  
Vol 8 (1) ◽  
Author(s):  
Ryan Joseph Rogers ◽  
Ning Zhong
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Euclidean Spaces ◽  
Definition Of

In this note, we provide the definition of a metric space and establish that, while all Euclidean spaces are metric spaces, not all metric spaces are Euclidean spaces. It is then natural and interesting to ask which theorems that hold in Euclidean spaces can be extended to general metric spaces and which ones cannot be extended. We survey this topic by considering six well-known theorems which hold in Euclidean spaces and rigorously exploring their validities in general metric spaces.

scholarly journals Metric Spaces Admitting Low-distortion Embeddings into All n-dimensional Banach Spaces

2016 ◽  
Vol 68 (4) ◽  
pp. 876-907 ◽  
Author(s):  
Mikhail Ostrovskii ◽  
Beata Randrianantoanina
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Spaces ◽  
Euclidean Spaces ◽  
Dimensional Banach Space ◽  
Low Distortion ◽  
Bounded Distortion ◽  
Finite Metric Spaces ◽  
Uniformly Bounded

AbstractFor a fixed K > 1 and n ∈ ℕ, n ≫ 1, we study metric spaces which admit embeddings with distortion ≤ K into each n-dimensional Banach space. Classical examples include spaces embeddable into log n-dimensional Euclidean spaces, and equilateral spaces.We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that n-point ultrametrics can be embedded with uniformly bounded distortions into arbitrary Banach spaces of dimension log n.The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension n. This partially answers a question of G. Schechtman.

On computably locally compact Hausdorff spaces

2009 ◽  
Vol 19 (1) ◽  
pp. 101-117
Author(s):  
YATAO XU ◽  
TANJA GRUBBA
Keyword(s):  
Metric Spaces ◽  
Euclidean Spaces ◽  
Hausdorff Spaces ◽  
Specific Kind ◽  
Locally Compact ◽  
Complete Investigation ◽  
Infinite Sequences ◽  
Compact Subsets

Locally compact Hausdorff spaces generalise Euclidean spaces and metric spaces from ‘metric’ to ‘topology’. But does the effectivity on the latter (Brattka and Weihrauch 1999; Weihrauch 2000) still hold for the former? In fact, some results will be totally changed. This paper provides a complete investigation of a specific kind of space – computably locally compact Hausdorff spaces. First we characterise this type of effective space, and then study computability on closed and compact subsets of them. We use the framework of the representation approach, TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notations and representations.

scholarly journals On the Product of Separable Metric Spaces

2001 ◽  
Vol 8 (4) ◽  
pp. 785-790
Author(s):  
D. Kighuradze
Keyword(s):  
Metric Spaces ◽  
Geometric Construction ◽  
Dimension Function ◽  
Euclidean Spaces

Abstract Some properties of the dimension function dim on the class of separable metric spaces are studied by means of geometric construction which can be realized in Euclidean spaces. In particular, we prove that if dim(𝑋 × 𝙔) = dim 𝑋 + dim 𝙔 for separable metric spaces 𝑋 and 𝙔, then there exists a pair of maps , , 𝑠 = dim 𝑋 + dim 𝙔, with stable intersections.

scholarly journals FINITE FLAT SPACES

Mathematika ◽  
2019 ◽  
Vol 65 (4) ◽  
pp. 1010-1017
Author(s):  
Vladimir Zolotov
Keyword(s):  
Lie Groups ◽  
Metric Spaces ◽  
List Type ◽  
Euclidean Spaces ◽  
Markov Type ◽  
Finite Metric Space ◽  
Isometric Actions ◽  
Compact Flat Manifolds ◽  
Flat Manifolds

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$ if for every $\unicode[STIX]{x1D716}>0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less than $1+\unicode[STIX]{x1D716}$. We show that almost isometric embeddability conditions are equal for the following classes of spaces.(a)Quotients of Euclidean spaces by isometric actions of finite groups.(b)$L_{2}$-Wasserstein spaces over Euclidean spaces.(c)Compact flat manifolds.(d)Compact flat orbifolds.(e)Quotients of connected compact bi-invariant Lie groups by isometric actions of compact Lie groups. (This one is the most surprising.)We call spaces which satisfy these conditions finite flat spaces. Since Markov-type constants depend only on finite subsets, we can conclude that connected compact bi-invariant Lie groups and their quotients have Markov type 2 with constant 1.

scholarly journals Qualitative Spatial Logic over 2D Euclidean Spaces Is Not Finitely Axiomatisable

2019 ◽  
Vol 33 ◽  
pp. 2776-2783
Author(s):  
Heshan Du ◽  
Natasha Alechina
Keyword(s):  
Metric Spaces ◽  
Geospatial Data ◽  
Euclidean Spaces ◽  
Spatial Logic ◽  
Spatial Logics

Several qualitative spatial logics used in reasoning about geospatial data have a sound and complete axiomatisation over metric spaces. It has been open whether the same axiomatisation is also sound and complete for 2D Euclidean spaces. We answer this question negatively by showing that the axiomatisations presented in (Du et al. 2013; Du and Alechina 2016) are not complete for 2D Euclidean spaces and, moreover, the logics are not finitely axiomatisable.

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