Geodesic convexity

<h2 id="definitions">Definitions</h2>
Let (M, g) be a Riemannian manifold.

<ul><li>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 <a href="/facts/Geodesic/fbWR6l80">geodesic</a> contained within C that joins those two points.</li>
<li>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</li></ul>

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.
<h2 id="properties">Properties</h2>
<ul><li>A geodesically convex (subset of a) Riemannian manifold is also a <a href="/facts/Convex_metric_space/YN68W4fe">convex metric space</a> with respect to the geodesic distance.</li></ul>
<h2 id="examples">Examples</h2>
<ul><li>A subset of n-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> En with its usual flat metric is geodesically convex <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it is convex in the usual sense, and similarly for functions.</li>
<li>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 <a href="/facts/Latitude/IBYqkjhU">latitude</a> further north than 45° south is not geodesically convex, since the minimizing geodesic (<a href="/facts/Great_circle/x4fq6eNm">great circle</a>) arc joining two distinct points on the southern boundary of A leaves A (e.g. in the case of two points 180° apart in <a href="/facts/Longitude/HsC61ML9">longitude</a>, the geodesic arc passes over the south pole).</li></ul>

<ul><li>Rapcsák, Tamás (1997). Smooth nonlinear optimization in Rn. Nonconvex Optimization and its Applications. Vol. 19. Dordrecht: Kluwer Academic Publishers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7923-4680-7. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1480415">1480415</a>.</li></ul>
<ul><li>Udriste, Constantin (1994). Convex functions and optimization methods on Riemannian manifolds. Mathematics and its Applications. Vol. 297. Dordrecht: Kluwer Academic Publishers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7923-3002-1.</li></ul>

Geodesic convexity open-in-new

Geodesic convexity