This question is motivated by the post Uncountable intersections of open balls in a separable metric space.
The general problem is the following: given a connected Riemannian manifold $M$, what are the sets that can be written as an (not necessarily countable) intersection of open balls?
If $M$ is a round sphere, then any subset is an intersection of open balls. If $M$ is a flat euclidean space, then my guess is that sets that are intersection of open balls are precisely the bounded convex sets. This seems to be related to the cut locus of points.
A specific question is:
What are the compact connected Riemannian manifolds on which all sets are intersection of open balls?