Is every closed and g-convex set topologically equivalent to a closed ball?

Question

Let $A \subset M$ be a closed and g-convex (geodesically convex) set on a Riemannian and compact and geodesically complete manifold $M$. It is known that when $M$ is the euclidean space $A$ is a topological closed ball.

I wonder if every closed and g-convex set is topologically equivalent to a closed ball.

Thanks in advance.

What is your definition of geodesic convexity? You should also be asking about a Riemannian manifold, not just a smooth manifold. — Anthony Carapetis, Mar 29 '18 at 07:26
I replaced the statement of smooth manifold by Riemannian and compact manifoldl — jaogye, Mar 29 '18 at 07:36
And g-convexity? Do you just mean that any pair of points $p,q \in A$ can be joined by a minimizing geodesic? Must this geodesic be unique? Must it lie in $A$? — Anthony Carapetis, Mar 29 '18 at 07:52
If that's the case then $M$ itself (assuming connected) is a counterxample, right? — freakish, Mar 29 '18 at 07:55
In order to be more precise I added the condition geodesically complete for $M$. — jaogye, Mar 29 '18 at 09:14

score 2 · Accepted Answer · answered Mar 29 '18 at 08:18

If $A=\bar S$ for an open set $S$ and any two points in $A$ can be joined by a unique minimizing geodesic with interior completely lying in $S$ (i.e. $S$ is strongly convex), then $S$ is homeomorphic to the open unit ball by this argument. This does not directly imply that $A$ is homeomorphic to a closed unit ball (think of the sphere with a point removed - this is topologically an open disk, but its closure in the sphere is not a closed disk). However, one could think of finding $S'\supset A $ also strongly convex and using a homeomorphism of $S'$ to a larger ball, but apparently this does not always work, see here.

If you do not assume that $A$ is the closure of an open, strongly convex set, then there are many counterexamples, as has been pointed out in the comments already. The most apparent being $A=M$ itself or $A$ being a small closed segment of a geodesic.

The set $A$ that I am thinking is the level set of a g-convex function $f:M \rightarrow \mathbb{R}$. So you answer can work since I am interested to see if $A$ contains holes or not. Dear @Jan Bohr many thanks for your answer. — jaogye, Mar 29 '18 at 09:51

Is every closed and g-convex set topologically equivalent to a closed ball?

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