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.
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.