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 (X, d) 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 (X, d) 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: does not imply convexity in the usual sense for subsets of Euclidean space (see the example of the rational numbers) nor does it imply path-connectedness (see the example of the rational numbers) nor does it imply geodesic convexity for Riemannian manifolds (consider, for example, the Euclidean plane with a closed disc removed). Euclidean spaces, that is, the usual three-dimensional space and its analogues for other dimensions, are convex metric spaces. Given any two distinct points and in such a space, the set of all points satisfying the above "triangle equality" forms the line segment between and which always has other points except and in fact, it has a continuum of points. Any 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 (this is a particular case of a more general statement to be discussed below). A circle is a convex metric space, if the distance between two points is defined as the length of the shortest arc on the circle connecting them. Let be a metric space (which is not necessarily convex). 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.