A topological space is weakly contractible if all the homotopy groups are trivial.
It's connected if it's not the union of two disjoint nonempty open sets.
A metric space $(X,d)$ is uniquely geodesic if two points $x,y \in X$ are connected by a unique path of minimal length, precisely $d(x,y)$.
Question : Is a weakly contractible connected metric space, uniquely geodesic ?
In the case of a negative answer :
- What are the classical counter-examples ?
- Are there natural additive conditions for having an affirmative answer ?