Talk:Convex metric space

From Wikipedia, the free encyclopedia

[edit] 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)

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)

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)

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