**Question:** Is every open convex subset $C$ of a Riemannian manifold $M$, necessarily contractible?

Here by a "convex subset" I mean a set $C$ having the property that between each pair of points in $C$ there is a unique geodesic contained in $C$. (For example an open hemisphere is convex inside the sphere)

I have tried to construct a deformation retract along the unique geodesic connecting every point in $C$ to a fixed point in $C$. But is this map always continuous? If the answer is not positive in general, I'm also interested in the special case of complete manifolds with non-positive curvature.

**Added.** For the case of complete manifolds with non-positive curvature one can argue as follows: Let's $q\in C$ be the fixed point and $p\in C$. Now if $\gamma(t) = \mathrm{exp}(tv) (t\in[0,1])$ is the unique geodesic in $C$ connecting $q$ to $p$, there is a neighborhood $U$ of $v$ in $T_q M$ s.t. $\exp([0,1]*U)$ is contained in $C$. Now since $\exp$ has no critical points in non-positive curvature, $\exp(U)$ contains an open neighborhood of $q$ and we can use inverse mapping theorem.

Thanks!