Geodesic convexity

Definitions

• 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 is said to be a geodesically convex function if the composition

  Properties

• A geodesically convex 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 arc joining two distinct points on the southern boundary of A leaves A.