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.

## Contents

* 1 Definitions
* 2 Properties
* 3 Examples
* 4 References

## 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 is said to be a (strictly) geodesically convex function if the composition

f∘γ:[0,T]->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. 19. Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-4680-7.

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

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