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 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).

scholarly journals AN EXAMPLE CONCERNING BOUNDED LINEAR REGULARITY OF SUBSPACES IN HILBERT SPACE

2013 ◽  
Vol 89 (2) ◽  
pp. 217-226 ◽  
Author(s):  
SIMEON REICH ◽  
ALEXANDER J. ZASLAVSKI
Keyword(s):  
Hilbert Space ◽  
Natural Number ◽  
Metric Spaces ◽  
Regularity Property ◽  
Finite Sets ◽  
Bounded Linear ◽  
Linear Regularity ◽  
Related Theorem

AbstractWe study bounded linear regularity of finite sets of closed subspaces in a Hilbert space. In particular, we construct for each natural number $n\geq 3$ a set of $n$ closed subspaces of ${\ell }^{2} $ which has the bounded linear regularity property, while the bounded linear regularity property does not hold for each one of its nonempty, proper nonsingleton subsets. We also establish a related theorem regarding the bounded regularity property in metric spaces.

EMBEDDABILITY OF GENERALISED WREATH PRODUCTS

2014 ◽  
Vol 91 (2) ◽  
pp. 250-263 ◽  
Author(s):  
CHRIS CAVE ◽  
DENNIS DREESEN
Keyword(s):  
Hilbert Space ◽  
Wreath Product ◽  
Metric Spaces ◽  
Wreath Products ◽  
Finitely Generated ◽  
Product Construction ◽  
Finitely Generated Groups

AbstractGiven two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces $X,Y,Z$ and derive a condition, called the (${\it\delta}$-polynomial) path lifting property, such that coarse embeddability of $X,Y$ and $Z$ implies coarse embeddability of $X\wr _{Z}Y$. We also give bounds on the compression of $X\wr _{Z}Y$ in terms of ${\it\delta}$ and the compressions of $X,Y$ and $Z$.

Estimation on Metric Spaces with Photometric Variation

2006 ◽  
Author(s):  
Ulf Grenander ◽  
Michael I. Miller
Keyword(s):  
Hilbert Space ◽  
Model Uncertainty ◽  
Metric Spaces ◽  
Basis Expansion ◽  
Image Field ◽  
Deformable Template ◽  
Photometric Variation

Model uncertainty comes in many forms. In this chapterweshall examine extensively the variable photometric model in which the underlying image field I(·) is modelled as an element of a Hilbert space I(·) ∈ H(φ) constructed via basis expansion {φi }. Inference involves both the pose and identity of the objects as well as the photometric intensity itself. This corresponds to making the template random, expanding the deformable template to include both the photometric variations and geometric variations.

scholarly journals Warped cones over profinite completions

2018 ◽  
Vol 10 (03) ◽  
pp. 563-584 ◽  
Author(s):  
Damian Sawicki
Keyword(s):  
Hilbert Space ◽  
Metric Spaces ◽  
Profinite Completion ◽  
Property A ◽  
Profinite Completions ◽  
The Way

We construct metric spaces that do not have property A yet are coarsely embeddable into the Hilbert space. Our examples are so-called warped cones, which were introduced by J. Roe to serve as examples of spaces non-embeddable into a Hilbert space and with or without property A. The construction provides the first examples of warped cones combining coarse embeddability and lack of property A. We also construct warped cones over manifolds with isometrically embedded expanders and generalise Roe’s criteria for the lack of property A or coarse embeddability of a warped cone. Along the way, it is proven that property A of the warped cone over a profinite completion is equivalent to amenability of the group. In the Appendix we solve a problem of Nowak regarding his examples of spaces with similar properties.

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