mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

— specifically, in

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

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

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

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 locally 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 conn ...

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 ::

f \circ \gamma :

, T The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

\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 In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints. Formally, consider a metric space (''X'', ''d'') and let ''x ...

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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

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 geographic coordinate system, geographic 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 t ...

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. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...

) 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 lett ...

, 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)