Geodesic convexity explained

In mathematics - specifically, in Riemannian geometry - 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 R is said to be a (strictly) geodesically convex function if the compositionf \circ \gamma : [ 0 , T] \to R

is a (strictly) convex function in the usual sense for every unit speed geodesic arc γ : [0, ''T''] → 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).

References

• Book: Rapcsák , Tamás . Smooth nonlinear optimization in Rⁿ. Nonconvex Optimization and its Applications . 19. Kluwer Academic Publishers. Dordrecht. 1997. 0-7923-4680-7. 1480415.
• Book: Udriste , Constantin . Convex functions and optimization methods on Riemannian manifolds. Mathematics and its Applications . 297. Kluwer Academic Publishers. Dordrecht. 1994. 0-7923-3002-1.