There is a concept 'convex neighborhood' that appear in do carmo's riemannian geometry, and he use this concept in the text and the Exercise3.4.
What confused me is that he had never given the formal definition of it. He only defined a subset $S$ to be 'strongly convex' If for any two point in the closure $\overline S$ there exists unique minimizing geodesic $\gamma$ joining them whose interior is contained in $S$.
Then I thought, well, maybe 'convex neighborhood' is just a neighborhood that is strongly convex, but then I have a question about the definition of 'strongly convex':
Does it mean that for any two point in the closure $\overline S$ there exists unique minimizing geodesic $\gamma$ joining them in the entire manifold $M$ and with the property that it's interior is contained in $S$, or does it mean that there maybe many minimizing geodesic $\gamma$ joining those two points in the entire manifold $M$, but only one has it's interior contained in $S$?
This really confused me, and then I found another book in which defined convex set in a riemannian manifold as follows: A subset $S$ of a riemannian manifold $M$ is 'convex' If for any two point in $S$ there exists minimizing geodesic $\gamma$ joining them in $M$, and any such minimizing geodesic join those two points is contained in $S$.
Can anyone tell me what is the common meaning of 'convex neighborhood'? Any help will be appreciated.