Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact).
Question 1. Is it true that every point of $X$ has a neighborhood $U$ such that any two points from $U$ can be connected by at most one shortest path (which does not have to be contained in $U$)?
Question 2. In the previous question, can one choose $U$ to be geodesically convex, i.e. for any two points from $U$ any shortest path between them (if it is not unique) is contained in $U$?
A reference would be helpful.
Remark. Of course, if $X$ is a smooth Riemannian manifold then the answers to both questions are 'yes'.