In mathematics — specifically, in

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to po ...

— geodesic convexity is a natural generalization of convexity for sets and functions to

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...

s. 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 In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

contained within ''C'' that joins those two points. * Let ''C'' be a geodesically convex subset of ''M''. A function

f:C\to\mathbf

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

\to \mathbf

: is a (strictly) convex function in the usual sense for every unit speed geodesic arc ''γ'' : , ''T''nbsp;→ ''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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

E^''n'' with its usual flat metric is geodesically convex

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

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 In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north po ...

further north than 45° south is ''not'' geodesically convex, since the minimizing geodesic (

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry ...

) 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 Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...

, the geodesic arc passes over the south pole).

References

* * {{cite book , last = Udriste , first = Constantin , title = Convex functions and optimization methods on Riemannian manifolds , series = Mathematics and its Applications , volume = 297 , publisher = Kluwer Academic Publishers , location = Dordrecht , year = 1994 , isbn = 0-7923-3002-1 Convex optimization Riemannian manifolds Geodesic (mathematics)