Examples
- 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.
Read more about this topic: Convex Metric Space
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