Question

Suppose $(M,g)$ is an open complete nonnegatively curved Riemannian manifold with $d$ its distance.
A totally convex set $C\subset M$ has the property that for any two point $x,y∈C$ $any$ geodesic (not only the minimal ones) joining them must lie in $C$.
If $C$ is totally convex, the fattened set $C^a=\{x\in M \, | \, d(x,C)\leq a\}$ is still totally convex? Is it at least for small $a$?

This question is somehow related to question "Examples on small cut radius of totally convex set in non-negatively curved manifold", http://mathoverflow.net/questions/119258

Answer 1

No. Consider a rotation-symmetric metric on $\mathbb R^2$ resembling a small spherical cap extented by a flat cone. A sufficiently short geodesic segment at the origin is totally convex in your sense. But a small neighborhood of such a segment is not convex due to positive curvature.