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Is there a term for a metric space that satisfies the following condition:

There exists a $C \geq 0$ such that for all $x,y \in X$ there exists a path $\gamma :[0,1] \to X$ with $\gamma(0)=x$ and $\gamma(1) = y$ such that $$l(\gamma) \leq C \cdot d(x,y). $$

So what I want is a space that is almost geodesic. Respectively where there are paths that are not too long compared to an ideal geodesic.

Loreno Heer
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    Think of $X= \mathbb R^2 \setminus { (x,0): x<0}$ with the standard metric (that on $\mathbb R^2$. Then there isn't such a $C$. (On the other hand, if you have a connected Riemannian manifold with the metric defined on the Riemannian structure, then such a $C$ can be found (this is almost by definition). –  Jun 23 '15 at 22:53
  • @John I am not sure I understand your example. How does that show that there could not be spaces where there is such a $C$? – Loreno Heer Jun 23 '15 at 22:56
  • @John I mean my question is, is there a name for a space for which there exists such a $C$? – Loreno Heer Jun 23 '15 at 22:57
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    I can only tell what kinds of metric spaces might satisfies your condition. But i am not sure if there is a name for this though (Also you may take a look of "length space" in metric geoemtry) –  Jun 23 '15 at 23:03

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Of course $C\ge1$ here. The standard term for such spaces is quasiconvex; search for "quasiconvex metric space" to find examples of usage.

The special case $C=1$ is a geodesic space. If the property holds for every $C>1$ (but not necessarily for $1$), this is a length space.