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.

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 ${\displaystyle f:C\to \mathbf {R} }$ is said to be a (strictly) geodesically convex function if the composition

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

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

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

• Udriste, Constantin (1994). Convex functions and optimization methods on Riemannian manifolds. Mathematics and its Applications. Vol. 297. Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-3002-1.