Here's another question that came to mind when I was reading the article on convex metric spaces in Wikipedia:

According to the article, "a circle, with the distance between two points measured along the shortest arc connecting them, is a (complete) convex metric space. Yet, if $x$ and $y$ are two points on a circle diametrically opposite to each other, there exist two metric segments conecting them (the two arcs into which these points split the circle), and those two arcs are metrically convex, but their intersection is the set $\{x,y\}$ which is not metrically convex."

In the article, a circle is treated as closed subset of Euclidean space, and according to another part of the article, "[a]ny convex set in a Euclidean space is a convex metric space with the induced Euclidean norm. For closed sets the converse is also true: if a closed subset of a Euclidean space together with the induced distance is a convex metric space, then it is a convex set."

If this is true, that is, if a circle is a closed subset of Euclidean space with an induced norm (the length of a segment along the shortest path between any two points on the circle) and is a convex metric space, being therefore a convex set, why isn't the intersection $\{x,y\}$ also metrically convex? Are the points $\{x,y\}$ as a subset of the circle not metrically convex on their own?

## Best Answer

As you said "a circle, with the distance between two points measured along the shortest arc connecting them, is a (complete) convex metric space." But its not a convex metric space with the norm induced by Euclidean norm. So the statement "if a closed subset of a Euclidean space together with the induced distance is a convex metric space, then it is a convex set" is not satisfying.