Geodesic convexity

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.

[edit] 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 geodesic arc contained within C that joins those two points.

Let C be a geodesically convex subset of M. A function f : C → R is said to be a (strictly) geodesically convex function if the composition

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

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

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

A subset of n-dimensional Euclidean space Eⁿ with its usual flat metric is 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 geodesic (great circle) joining two points on the southern boundary of A may well leave A (e.g. in the case of two points 180° apart in longitude, in which case the geodesic arc passes over the south pole).

Rapcsák, Tamás (1997). Smooth nonlinear optimization in Rⁿ, Nonconvex Optimization and its Applications 19. Dordrecht: Kluwer Academic Publishers, xiv+374. ISBN 0-7923-4680-7. MR 1480415