Talk:Convex metric space

Metric and usual convexity

I think that we need to further stress the difference between, metric convexity and usual convexity, in that some sets that are non-convex in the usual sense are convex by this article's definition. For example, let (X, d) be the metric space X = R \ [−1, +1] with d the usual Euclidean distance inherited from R. Then X is convex, since given any two points x and y in X, there is some z in X that is between them: if x and y lie on the same side of the "hole" [−1, +1], simply take z = (x + y) / 2; otherwise, assume without loss of generality that x < −1 and take z = (x −1) / 2. Sullivan.t.j 10:04, 9 August 2007 (UTC)[reply]

Example 3 in the article shows that a circle, while not a convex set, is a convex metric space. I see you are suggesting another such example above. Feel free to add it in. One could also put in a blurb in the intro saying that convex metric spaces are not necessarily convex sets. Oleg Alexandrov (talk) 16:23, 9 August 2007 (UTC)[reply]

I will add such a caveat as you suggest. However, my point goes even further. Not only does metric convexity not imply usual convexity (as in the example of the circle), it does not imply geodesic convexity (e.g. the line with a closed interval removed, or the plane with a closed disc removed). Sullivan.t.j 12:25, 10 August 2007 (UTC)[reply]

What do you mean by a line with a closed interval removed? If you restrict the distance function on line to this subset, then it is not convex, and so how does this give contradiction to claim that convexity implies geodesic convexity? Lost-n-translation (talk) 23:09, 5 December 2017 (UTC)[reply]

Thanks for the clarification in the article. Oleg Alexandrov (talk) 05:14, 11 August 2007 (UTC)[reply]

Convex metric space is old name geodesic space. In the article a wrong def is given. Why revert my edit...--Tosha (talk) 15:03, 9 September 2008 (UTC)[reply]

Your edit was reverted because the definitions given in the two articles define different concepts. Sullivan.t.j (talk) 23:16, 9 September 2008 (UTC)[reply]

There are three standard defs which reflect different level of the same thing:

intrinsic metric= existance of almost midpoint (the def in Intrinsic metric is wrong)
lenght metric = distance is infimim of lengths of paths connecting two points (that is what called Intrinsic metric in Intrinsic metric)
geodesic metric = any two points can be joined by minimizing geodesic (that is also convex metric space in old german way, the definition here is wrong)

So all this should be in one place... --Tosha (talk) 12:37, 12 September 2008 (UTC)[reply]

The first two are only equivalent for complete metric spaces. The second two are only equivalent for complete and locally compact metric spaces; see Hopf-Rinow theorem. There are counterexamples:

The rational numbers with the induced Euclidean metric are convex but not a length metric space.
The plane with a bounded open line segment removed are a length metric space which is not convex.
There is an example due to C.J. Atkin (referenced in Hopf-Rinow theorem) of a non-locally compact complete intrinsic metric space in which two points are not joined by a geodesic.

--siℓℓy rabbit (talk) 12:36, 15 September 2008 (UTC)[reply]