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

f ∘ γ : [ 0 , T ] → R {\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).

• 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.