Talk:Geodesic convexity

Relation to convex metric space
I realize that Riemannian metrics are not the same thing as distance metrics, but the definition of convexity here is almost the same as that in convex metric space (or rather, not the main definition there, but the related one mentioned in the article and used more explicitly e.g. in injective metric space that any pair of points can be connected by the isometric image of a line segment). Should there be a closer connection between these articles? Merge, maybe? Or rename this article geodesic metric space, discuss in it non-Riemannian as well as Riemannian metrics, clarify the difference between that and the notion of convexity discussed in convex metric space, and link articles that need this version of the concept, such as injective metric space, to here instead of there? —David Eppstein 17:09, 8 August 2007 (UTC)


 * I agree wholeheartedly. Having a page devoted to the property of "Geodesic convexity" rather than about "Spaces with convex geodesics" is the problem.  The argument in the "disputed" section below seems to be an artifact of trying to use a particular definition of convexity without having nailed down the space we're talking about (i.e. Riemannian manifold or length metric). Tom3118 (talk) 13:30, 8 September 2010 (UTC)

Disputed
Where does the quoted definition come from?

It does not say that it is required for the geodesic to be minimizing the distance between the two given points (i.e. the length of the arc is required to be the distance between the two points). It is not mentioned if other geodesics (minimizing or not) are allowed connecting the two points, and if allowed, if these must also lie completely within C.

The example with the "big" disc A inside S2 only makes sense if one requires the geodesics to be minimizers.

In the book, three distinct definitions are given:


 * weakly convex: for any two points from C there exists exactly one minimizing geodesic in C connecting them
 * convex: for any two points in C there exists exactly one minimizing geodesic in M connecting them, and that geodesic arc lies completely in C
 * strongly convex: for any two points in C there exists exactly one minimizing geodesic in M connecting them, and that geodesic arc lies completely in C; and furthermore there exists no nonminimizing geodesic inside C connecting the two points.

Consider the open northern hemisphere of S2. It is strongly convex. Now add two antipodal points from the equator. The resulting set is only weakly convex. The closed northern hemisphere (whole equator included) is not even weakly convex.

Other definitions are possible.

/84.238.115.151 (talk) 00:06, 15 April 2009 (UTC)


 * Sorry, the upper hemispere plus two antipodal equatorial points is not weakly convex in this sense. Consider instead for example half a great circle including endpoints. /84.238.115.151 (talk) 20:50, 15 April 2009 (UTC)


 * I'm assuming that if A and B are geodesically convex, we want their intersection to be geodesically convex as well. That means that if p and q are points in the intersection of A and B, I would expect the set of geodesics connecting p to q in A to be the same as the set of geodesics connecting p and q in B, so that when we intersect A and B that same set of geodesics is contained in the result, making it geodesically convex. That is not the case with the current definition on the page, so it apparently violates the principle of preservation of convexity under intersection. Probably the best way to fix that is to do as suggested above and require that the geodesics to be length minimizers. In certain cases it might work to define a set as geodesically convex if and only if for every two points it contains, it contains every geodesic between them - that would also lead to a geodesic convexity preserved under intersection. —Preceding unsigned comment added by 128.61.118.127 (talk) 15:11, 23 November 2009 (UTC)


 * I don't think it is a good idea to require a convex set to contain every geodesic between two of its points. for example, on a sphere it will mean that every convex set with at least two points have to be the whole sphere. 79.178.51.153 (talk) 15:01, 24 December 2010 (UTC)


 * There are even weaker versions of convexity, only requiring the following:
 * for any two points from C if there exists exactly one minimizing geodesic between them, it must be contained in C.
 * As an example where such notion appears, see for the case of the sphere.160.45.109.217 (talk) 17:33, 8 November 2012 (UTC)


 * I ran a quick survey of some common differential geometry textbooks, to try to see the consensus of the literature. Here's the results:
 * p. 149: "$&delta;$ is convex in the sense that any two points of $q_{1}$ can be joined by a geodesic which lies in $q_{2}$." p. 166: "We now proceed to prove the existence of a convex neighborhood around each point of a Riemannian manifold in the following form....(1) Any two points of $A&subseteq;M$ can be joined by a unique minimizing geodesic; and it is the unique geodesic joining the two points and lying in $&part;A$; (2) In $1$, the square of the distance $U&subseteq;M$ is a differentiable function of $W$ and $W$."
 * I conclude that the article does in fact use the correct definition. Bernanke&#39;s Crossbow (talk) 21:00, 15 September 2022 (UTC)
 * p. 149: "$q,q&prime;&isin;U$ is convex in the sense that any two points of $U(x_{0};&rho;)$ can be joined by a geodesic which lies in $U(x_{0};&rho;)$." p. 166: "We now proceed to prove the existence of a convex neighborhood around each point of a Riemannian manifold in the following form....(1) Any two points of $U(x_{0};&rho;)$ can be joined by a unique minimizing geodesic; and it is the unique geodesic joining the two points and lying in $U(x;&rho;)$; (2) In $U(x;&rho;)$, the square of the distance $U(x;&rho;)$ is a differentiable function of $W$ and $W$."
 * I conclude that the article does in fact use the correct definition. Bernanke&#39;s Crossbow (talk) 21:00, 15 September 2022 (UTC)
 * p. 149: "$d(y,z)$ is convex in the sense that any two points of $U&subseteq;M$ can be joined by a geodesic which lies in $\mathcal{N}_{p}$." p. 166: "We now proceed to prove the existence of a convex neighborhood around each point of a Riemannian manifold in the following form....(1) Any two points of $\mathcal{N}_{p}$ can be joined by a unique minimizing geodesic; and it is the unique geodesic joining the two points and lying in $\mathcal{N}_{p}$; (2) In $\mathcal{N}_{p}$, the square of the distance ᙭᙭᙭ is a differentiable function of $M$ and $A$."
 * I conclude that the article does in fact use the correct definition. Bernanke&#39;s Crossbow (talk) 21:00, 15 September 2022 (UTC)
 * I conclude that the article does in fact use the correct definition. Bernanke&#39;s Crossbow (talk) 21:00, 15 September 2022 (UTC)
 * I conclude that the article does in fact use the correct definition. Bernanke&#39;s Crossbow (talk) 21:00, 15 September 2022 (UTC)