Convex metric space


In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints.
Formally, consider a metric space and let x and y be two points in X. A point z in X is said to be between x and y if all three points are distinct, and
that is, the triangle inequality becomes an equality. A convex metric space is a metric space such that, for any two distinct points x and y in X, there exists a third point z in X lying between x and y.
Metric convexity:
Let be a metric space. A subset of is called a metric segment between two distinct points and in if there exists a closed interval on the real line and an isometry
such that and
It is clear that any point in such a metric segment except for the "endpoints" and is between and As such, if a metric space admits metric segments between any two distinct points in the space, then it is a convex metric space.
The converse is not true, in general. The rational numbers form a convex metric space with the usual distance, yet there exists no segment connecting two rational numbers which is made up of rational numbers only. If however, is a convex metric space, and, in addition, it is complete, one can prove that for any two points in there exists a metric segment connecting them .

Convex metric spaces and convex sets

As mentioned in the examples section, closed subsets of Euclidean spaces are convex metric spaces if and only if they are convex sets. It is then natural to think of convex metric spaces as generalizing the notion of convexity beyond Euclidean spaces, with usual linear segments replaced by metric segments.
It is important to note, however, that metric convexity defined this way does not have one of the most important properties of Euclidean convex sets, that being that the intersection of two convex sets is convex. Indeed, as mentioned in the examples section, a circle, with the distance between two points measured along the shortest arc connecting them, is a convex metric space. Yet, if and are two points on a circle diametrically opposite to each other, there exist two metric segments connecting them, and those two arcs are metrically convex, but their intersection is the set which is not metrically convex.