On Certain Metric Spaces Arising From Euclidean Spaces by a Change of Metric and Their Imbedding in Hilbert Space

1937 ◽  
Vol 38 (4) ◽  
pp. 787 ◽  
Author(s):  
I. J. Schoenberg
Keyword(s):  
Hilbert Space ◽  
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.

Embedding quasi-metric spaces in hilbert space

1981 ◽  
Vol 23 (1) ◽  
pp. 315-317
Author(s):  
Pham Son
Keyword(s):  
Hilbert Space ◽  
Metric Spaces

Positive definite functions on spheres

1973 ◽  
Vol 73 (1) ◽  
pp. 145-156 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Hilbert Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Distance ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Necessary Condition ◽  
Unit Hypersphere

Positive definite functions on metric spaces were considered by Schoenberg (26). We write σk for the unit hypersphere in (k + 1)-space; then σk is a metric space under geodesic distance. The functions which are positive definite (p.d.) on σk were characterized by Schoenberg (27), who also obtained a necessary condition for a function to be p.d. on the it sphere σ∞ in Hilbert space. We extend this result by showing that Schoenberg's necessary condition for a function to be p.d. on σ∞ is also sufficient.

scholarly journals Lipschitz Extensions to Finitely Many Points

2018 ◽  
Vol 6 (1) ◽  
pp. 174-191 ◽  
Author(s):  
Giuliano Basso
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Metric Spaces ◽  
Lipschitz Constant ◽  
Target Space ◽  
Square Root ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Lipschitz Extensions

AbstractWe consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.

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 Metric spaces which cannot be isometrically embedded in Hilbert space

1984 ◽  
Vol 30 (2) ◽  
pp. 161-167
Author(s):  
Yang Lu ◽  
Zhang Jing-Zhong
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Metric Spaces ◽  
Geometric Inequality ◽  
Cambridge Philos ◽  
Higher Dimensional ◽  
Planar Convex ◽  
Convex Quadrangle

Let A1A2A3A4, be a planar convex quadrangle with diagonals A1A3 and A2A4. Is there a quadrangle B1B2B3B4 in Euclidean space such that A1A3 < B1B3, A2A4 < B2B4 but AiAj > BiBj for other edges?The answer is “no”. It seems to be obvious but the proof is more difficult. In this paper we shall solve similar more complicated problems by using a higher dimensional geometric inequality which is a generalisation of the well-known Pedoe inequality (Proc. Cambridge Philos. Soc.38 (1942), 397–398) and an interesting result by L.M. Blumenthal and B.E. Gillam (Amer. Math. Monthly50 (1943), 181–185).

Sign in / Sign up

Export Citation Format

Share Document