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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) $d(x,y)+d(y,z) \geq d(x,z)$.
17
votes
1
answer
359
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Hopping geodesics
Is there a complete metric space $X$ with the following property?
For any pair of points $p,q\in X$ there is unique minimizing geodesic $[pq]_X$ that connects $p$ to $q$, but the map $(p,q)\mapsto …
- 45k
3
votes
1
answer
180
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Completeness of intrinsication
Lemma. Suppose $(X,\rho)$ is a complete metric space and $\hat \rho$ is its induced intrinsic metric. Then $(X,\hat \rho)$ is complete.
This lemma was essentially proved in [2.3. in Metric minimizing …
- 45k
6
votes
1
answer
233
views
m-point-homogeneous, but not (m+1)-point-homogeneous
It is straightforward to check that the discrete cube $Q=\{0,1\}^n$ with $\ell^1$-metric is 3-point-homogeneous, but not 4-point-homogeneous (assuming $n$ is large).
In other words, if $A\subset Q$ ha …
- 45k
9
votes
0
answers
482
views
Category of metric spaces
Is there a standard/good reference text that does category of metric spaces?
Say, it seems that by looking at this category one can recover everything about particular metric space up to scaling --- …
- 45k