High Quality Content by WIKIPEDIA articles!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.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 (complete) convex metric space. Yet, if x and y are two points on a circle diametrally opposite to each other, there exist two metric segments connecting 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.

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