I am trying to imagine normal balls which are not convex. Are there any nice exmaples?
Obviously, in the Euclidean space all normal balls are convex and it seems to me this is also true for the sphere.
Perhaps a space of negative curvature? (Is there a necessary condition on the curvature for the existence of non-convex normal balls?)
Let $M$ be a Riemannian manifold. A normal ball around $p$, is a set of the form of $\exp_p(B_r(0))$ where $\bar B_r(0)$ (the closed ball in $T_pM$ with radius $r$) is contained in an open set $V \subseteq T_pM$ such that $\exp_p:V \to \exp_p(V)$ is a diffeomorphism.
A subset $A \subseteq M$ is convex, if every two points in $A$ can be joined by a minimizing geodesic (Some people call this weak convexity I think, since I do not require uniqueness).