weakly geodesically convex if for any two points from there exists exactly one minimizing geodesic in connecting them;
geodesically convex if for any two points in there exists exactly one minimizing geodesic in connecting them, and that geodesic arc lies completely in
strongly geodesically convex if for any two points in there exists exactly one minimizing geodesic in connecting them, and that geodesic arc lies completely in ; and furthermore there exists no nonminimizing geodesic inside connecting the two points.
The convexity radius at a point is the supremum (which may be ) of such that for all the geodesic ball is strongly geodesically convex.
The convexity radius of is the infimum over the points of the convexity radii at these points.
For a metric space, a distance-preserving path is a function which is an isometry. This is a metric-space analogue of an “arc-length-parametrized minimizing geodesic” on a Riemannian manifold. In particular, the existence of such a path implies that . We then say that is geodesic (or geodesically convex) if any two points can be connected by a distance-preserving path.
We say that is a length space if for any and , the distance is the infimum of the lengths of all continuous paths from to . The Hopf-Rinow theorem? says that for a metric space in which every closed bounded subset is compact, being a length space is equivalent to being geodesic.
At any point of a Riemannian manifold, the convexity radius is positive.
This is due to (Whitehead).
If is a compact space then the convexity radius of is positive.
This is reproduced for instance as proposition 95 in (Berger)
This is shown in (Greene).
Original literature includes
A review of geodesic convexity in Riemannian manifolds is in
A categorical perspective on geodesic convexity for metric spaces can be found in
Much of the issues on geodesic convexity becomes more complicated when trying to generalize from Riemannian to Lorentzian manifolds, as discussed at length at
In particular, the conclusions of the Hopf-Rinow Theorem fail to hold for complete Lorentzian manifolds.