Convex Metric Space

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).