In the book Riemannian geometry - modern introduction by Isaac Chavel, three different definitions of convexity are introduced. I am looking for an example of a set which is convex but not strongly convex according to the following definitions (cited from the book):

Let $M$ be a complete Riemannian manifold, and $A \subset M$. $A$ is:

 

1. convex if for any $p,q\in A$ there exists a geodesic $\gamma_{pq} \subset A$ such that $\gamma_{pq}$ is the unique minimizer in $M$ connecting $p$ to $q$.
2. strongly convex if for any $p,q\in A$ there exists a geodesic $\gamma_{pq}\subset A$ such that $\gamma_{pq}$ is the unique minimizer in $M$ connecting $p$ to $q$, and $\gamma_{pq}$ is the only geodesic contained in $A$ joining $p$ and $q$.

I believe that this picture gives an example. This is not an accurate figure! The set I'm referring to is the one centered at around a geodesic segment with end points conjugate one to the other along it, and bounded symmetrically by two close non minimal geodesics.

Is my example correct? If not, then can someone provide a valid example?

Edit: Caution I know of something like 7-8 different definitions of convexity in the case of a Riemannian manifold. Try to refer to the definitions I gave above.

In the book Riemannian geometry - modern introduction by Isaac Chavel, three different definitions of convexity are introduced. I am looking for an example of a set which is convex but not strongly convex according to the following definitions (cited from the book):

Let $M$ be a complete Riemannian manifold, and $A \subset M$. $A$ is:

1. convex if for any $p,q\in A$ there exists a geodesic $\gamma_{pq} \subset A$ such that $\gamma_{pq}$ is the unique minimizer in $M$ connecting $p$ to $q$.
2. strongly convex if for any $p,q\in A$ there exists a geodesic $\gamma_{pq}\subset A$ such that $\gamma_{pq}$ is the unique minimizer in $M$ connecting $p$ to $q$, and $\gamma_{pq}$ is the only geodesic contained in $A$ joining $p$ and $q$.

I believe that this picture gives an example. This is not an accurate figure! The set I'm referring to is the one centered at around a geodesic segment with end points conjugate one to the other along it, and bounded symmetrically by two close non minimal geodesics.

Is my example correct? If not, then can someone provide a valid example?

Edit: Caution I know of something like 7-8 different definitions of convexity in the case of a Riemannian manifold. Try to refer to the definitions I gave above.

In the book Riemannian geometry - modern introduction by Isaac Chavel, three different definitions of convexity are introduced. I am looking for an example of a set which is convex but not strongly convex according to the following definitions (cited from the book):

Let $M$ be a complete Riemannian manifold, and $A \subset M$. $A$ is:

1. convex if for any $p,q\in A$ there exists a geodesic $\gamma_{pq} \subset A$ such that $\gamma_{pq}$ is the unique minimizer in $M$ connecting $p$ to $q$.
2. strongly convex if for any $p,q\in A$ there exists a geodesic $\gamma_{pq}\subset A$ such that $\gamma_{pq}$ is the unique minimizer in $M$ connecting $p$ to $q$, and $\gamma_{pq}$ is the only geodesic contained in $A$ joining $p$ and $q$.

I believe that this picture gives an example. This is not an accurate figure! The set I'm referring to is the one centered at around a geodesic segment with end points conjugate one to the other along it, and bounded symmetrically by two close non minimal geodesics.

Is my example correct? If not, then can someone provide a valid example?

In the book Riemannian geometry - modern introduction by Isaac Chavel, three different definitions of convexity are introduced. I am looking for an example of a set which is convex but not strongly convex according to the following definitions (cited from the book):

Let $M$ be a complete Riemannian manifold, and $A \subset M$. $A$ is:

1. convex if for any $p,q\in A$ there exists a geodesic $\gamma_{pq} \subset A$ such that $\gamma_{pq}$ is the unique minimizer in $M$ connecting $p$ to $q$.
2. strongly convex if for any $p,q\in A$ there exists a geodesic $\gamma_{pq}\subset A$ such that $\gamma_{pq}$ is the unique minimizer in $M$ connecting $p$ to $q$, and $\gamma_{pq}$ is the only geodesic contained in $A$ joining $p$ and $q$.

I believe that this picture gives an example. This is not an accurate figure! The set I'm referring to is the one centered at around a geodesic segment with end points conjugate one to the other along it, and bounded symmetrically by two close non minimal geodesics.

Is my example correct? If not, then can someone provide a valid example?

Edit: Caution I know of something like 7-8 different definitions of convexity in the case of a Riemannian manifold. Try to refer to the definitions I gave above.