TITLE: Uniquely geodesic and CAT(0) spaces?
QUESTION [6 upvotes]: Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.

Question : Is a finite dimensional metric space, uniquely geodesic if and only if it is CAT(0) ?

In the case of a negative answer :
- Is CAT(0) assumption necessary ?  Is it sufficient ?
- What are the classical counter-examples ?
- Is there a slight additive assumption for having a positive answer ?

REPLY [5 votes]: A good counterexample is the Teichmuller space of a closed oriented surface $S$. It is uniquely geodesic by Teichmuller's theorem, but it is not $CAT(0)$.