Boundedness of pretangent spaces to general metric spaces

@article{Bilet2012BoundednessOP,
  title={Boundedness of pretangent spaces to general metric spaces},
  author={Viktoriia Bilet and Oleksiy Dovgoshey},
  journal={arXiv: Metric Geometry},
  year={2012}
}
Let (X,d,p) be a metric space with a metric d and a marked point p. We define the set of w-strongly porous at 0 subsets of [0,\infty) and prove that the distance set {d(x,p): x\in X} is w-strongly porous at 0 if and only if every pretangent space to X at p is bounded. 
View PDF on arXiv
13 Citations
Background Citations
7

13 Citations

Uniform boundedness of pretangent spaces and local strong one-side porosity
Let (X,d,p) be a pointed metric space. A pretangent space to X at p is a metric space consisting of some equivalence classes of convergent to p sequences (x_n), x_n \in X, whose degree of convergence…
Completeness, Closedness and Metric Reflections of Pseudometric Spaces
. It is well-known that a metric space ( X, d ) is complete iff the set X is closed in every metric superspace of ( X, d ) . For a given pseudometric space ( Y, ρ ) , we describe the maximal class CEC…
On equivalence of unbounded metric spaces at infinity
Let (X, d) be an unbounded metric space. To investigate the asymptotic behavior of (X, d) at infinity, one can consider a sequence of rescaling metric spaces (X, 1 rn d) generated by given sequence…
ON THE ASYMPTOTIC EQUIVALENCE OF UNBOUNDED METRIC SPACES
Let (X, d) be an unbounded metric space. To investigate the asymptotic behavior of (X, d) at infinity, one can consider a sequence of rescaling metric spaces (X, 1 rn d) generated by given sequence…
Uniform boundedness of pretangent spaces, local constancy of metric derivatives and strong right upper porosity at a point
Let $(X,d,p)$ be a pointed metric space. A pretangent space to $X$ at $p$ is a metric space consisting of some equivalence classes of convergent to $p$ sequences $(x_n), x_n \in X,$ whose degree of…
Pretangent spaces with nonpositive and nonnegative Aleksandrov curvature
We find conditions under which the pretangent spaces to general metric spaces have the nonpositive Aleksandrov curvature or nonnegative one. The infinitesimal structure of general metric cpaces with…
C A ] 2 4 M ar 2 01 7 UNIONS AND IDEALS OF LOCALLY STRONGLY POROUS
For subsets of R = [0,∞) we introduce a notion of coherently porous sets as the sets for which the upper limit in the definition of porosity at a point is attained along the same sequence. We prove…
Finite Spaces Pretangent to Metric Spaces at Infinity
Let X be an unbounded metric space, and let \( \tilde{r} \) be a sequence of positive real numbers tending to infinity. We define the pretangent space \( {\Omega}_{\infty, \tilde{r}}^X \) to X at…
Local one-sided porosity and pretangent spaces
Abstract For subsets of ℝ + $\mathbb{R}^{+}$ we consider the local right upper porosity and the local right lower porosity as elements of a cluster set of all porosity numbers. The use of a scaling…
Asymptotic structure of general metric spaces at infinity
Let $(X,d)$ be an unbounded metric space and $\tilde r=(r_n)_{n\in\mathbb N}$ be a scaling sequence of positive real numbers tending to infinity. We define the pretangent space $\Omega_{\infty,…
...
...

References

SHOWING 1-10 OF 26 REFERENCES
Metric spaces with unique pretangent spaces
We find necessary and sufficient conditions under which an arbitrary metric space $X$ has a unique pretangent space at the marked point $a\in X$. Key words: Metric spaces; Tangent spaces to metric…
Compactness and Boundedness of Tangent Spaces to Metric Spaces
We describe metric spaces with bounded pretangent spaces and characterize proper metric spaces with proper tangent spaces. We also present the necessary and sufficient conditions under which a…
Sobolev spaces on an arbitrary metric space
We define Sobolev space W1,p for 1<p≤∞ on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev…
METRIC SPACES WITH UNIQUE PRETANGENT SPACES. CONDITIONS OF THE UNIQUENESS
We find necessary and sucient conditions for an arbitrary metric space X to have a unique pretangent space at a marked point a 2 X. Applying this general result we show that each logarithmic spiral…
Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
This paper studies a possible definition of Sobolev spaces in abstract metric spaces, and answers in the affirmative the question whether this definition yields a Banach space. The paper also…
Tangent spaces to metric spaces and to their subspaces
We investigate tangent spaces and metric space valued derivatives at points of a general metric spaces. The conditions under which two different subspaces of the metric space have isometric tangent…
Lectures on Analysis on Metric Spaces
1. Covering Theorems.- 2. Maximal Functions.- 3. Sobolev Spaces.- 4. Poincare Inequality.- 5. Sobolev Spaces on Metric Spaces.- 6. Lipschitz Functions.- 7. Modulus of a Curve Family, Capacity, and…
From local to global in quasiconformal structures.
  • J. Heinonen, P. Koskela
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1996
We exhibit a large class of metric spaces whose infinitesimal quasiconformal structure is strong enough to capture the global quasiconformal structure. A sufficient condition for this to happen is…
Differentiability of Lipschitz Functions on Metric Measure Spaces
Abstract. ((Without Abstract)).
A KIND OF LOCAL STRONG ONE-SIDE POROSITY
We define and study the completely strongly porous at 0 subsets of R^{+}. Several characterizations of these subsets are obtained, among them the description via an universal property and structural…
...
...