Geodesic convexity

In mathematics &mdash; specifically, in Riemannian geometry &mdash; geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds. It is common to drop the prefix "geodesic" and refer simply to "convexity" of a set or function.

Definitions
Let (M, g) be a Riemannian manifold.


 * A subset C of M is said to be a geodesically convex set if, given any two points in C, there is a unique minimizing geodesic contained within C that joins those two points.
 * Let C be a geodesically convex subset of M. A function $$f:C\to\mathbf{R}$$ is said to be a (strictly) geodesically convex function if the composition


 * $$f \circ \gamma : [0, T] \to \mathbf{R}$$


 * is a (strictly) convex function in the usual sense for every unit speed geodesic arc &gamma; : [0, T] &rarr; M contained within C.

Properties

 * A geodesically convex (subset of a) Riemannian manifold is also a convex metric space with respect to the geodesic distance.

Examples

 * A subset of n-dimensional Euclidean space En with its usual flat metric is geodesically convex if and only if it is convex in the usual sense, and similarly for functions.
 * The "northern hemisphere" of the 2-dimensional sphere S2 with its usual metric is geodesically convex. However, the subset A of S2 consisting of those points with latitude further north than 45° south is not geodesically convex, since the minimizing geodesic (great circle) arc joining two distinct points on the southern boundary of A leaves A (e.g. in the case of two points 180° apart in longitude, the geodesic arc passes over the south pole).