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Results tagged with metric-spaces
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user 1441
A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.
5
votes
Accepted
Independence of the axiomatics of metric cones
Yes, it is independent.
Consider function $f\colon \mathbb R^2\to \mathbb R_\ge$ defened the following way:
$$f(x,y)=f(y,x)$$
and if $|y|\le |x|$ then
$$f(x,y)=|x|+\min\{|y|,|y-\tfrac12{\cdot} x|\}.$ …
- 45k
6
votes
Accepted
Hopping geodesics
Yes, such examples do live. The following answer was given in "Metric spaces of non-positive curvature" by Bridson and Haefliger; thanks to GGT and Moishe Kohan.
- 45k
20
votes
Accepted
Which metric spaces have this superposition property?
If the metric space is locally compact and intrinsic, then you get only spheres, Euclidean spaces and hyperbolic spaces.
[See Metric methods in Finsler... by Busemann and Sur certaines classes d'espac …
- 45k
-1
votes
Hyperbolic space embeds into Wasserstein space
It admits an isometric embedding.
The Nash--Kuiper theorem provides an isometric embedding of $\mathbb{H}^n\to \mathbb{R}^{n+1}$; moreover, its image might be in a bounded set. Now observe that $\math …
- 45k
2
votes
Accepted
Hyperbolic space embeds into Wasserstein space
For global distortion.
Choose a 4 point set --- vertices of an equilateral triangle in $B(x,r)_{\mathbb{H}^n}$ and its center and observe that it cannot be emebdded isometrically in nonnegatively curv …
- 45k
2
votes
Accepted
If $X,X'$ have the same $\varepsilon$-packing numbers and $f:X \to X'$ surjective $1$-Lipsch...
Consider two metrics on $\{x,x',y\}$ defined by
$$|x-y|_1=|x'-y|_1=|x-y|_2=3, \quad |x-x'|_1=|x-x'|_2=1, \quad |x'-y|_2=2.$$
Denote by $X_1$ and $X_2$ the corresponding metric spaces.
Note that $\math …
- 45k
4
votes
Accepted
When are Wasserstein spaces $CAT(\kappa)$?
Almost never...
Note that there is an isometric embedding $X\to W_p(X)$, so $X$ has to be CAT(κ). Second the space $W_p(X)$ contains symmetric $p$-product $S^n(X)=X^{\times n}/S_n$ so $p=2$, or $X$ is …
- 45k
7
votes
Isometric embeddings of metric spaces in Hilbert spaces
This is the answer to the original question (Not the one which is posted now).
Look in two papers, mine and the paper of Enrico Le Donne.
You are looking for spaces which admit length-preserving e …
- 45k
4
votes
Accepted
Partitioning a smooth manifold into geodesically convex sets
Choose a triangulation of $X$.
Let us equip $X$ with a length metric such that each simplex is standard.
We may think that $X$ subcomplex of a standard simplex $S$ of large dimension.
Since each face …
- 45k
5
votes
Accepted
Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isome...
It follows from 1.6.15(1) in "A Course in Metric Geometry" by Burago, Burago, and Ivanov and 1.6.15(2) is a more general statement:
Any distance-noncontracting map from a compact metric space to itse …
- 45k