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 (strictly) geodesically convex function if the composition
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 spaceEn 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.
ISBN0-7923-4680-7.
MR1480415.
Udriste, Constantin (1994). Convex functions and optimization methods on Riemannian manifolds. Mathematics and its Applications. Vol. 297. Dordrecht: Kluwer Academic Publishers.
ISBN0-7923-3002-1.
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 (strictly) geodesically convex function if the composition
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 spaceEn 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.
ISBN0-7923-4680-7.
MR1480415.
Udriste, Constantin (1994). Convex functions and optimization methods on Riemannian manifolds. Mathematics and its Applications. Vol. 297. Dordrecht: Kluwer Academic Publishers.
ISBN0-7923-3002-1.