Examples for geodesic balls which are not convex

Question

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?)

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Reminders:

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).

Answer 1

(The question turned out to be rather easy, I am posting an answer based on Andrew D. Hwang's comments.

In the round unit sphere, a geodesic ball of radius greater than $\frac{\pi}{2}$ , which compactly contains a closed hemisphere, is not geodesically convex.

Also, as stated here, sufficiently small geodesic balls in a Riemannian manifold are convex.