  In this work we solve a couple of well known open problems related to the
quasihyperbolic metric. In the case of planar domains, our first main result
states that quasihyperbolic geodesics are unique in simply connected domains.
As the second main result, we prove that for an arbitrary plane domain the
geodesics are unique for all pairs of points with quasihyperbolic distance less
than $\pi$. This bound is sharp and improves the best known bound. Concerning
the $n$-dimensional case, we prove the existence of a universal constant $c$
(independent of dimension) such that any quasihyperbolic ball with radius less
than $c$ is convex.
