Properties
In an injective space, the radius of the minimum ball that contains any set S is equal to half the diameter of S. This follows since the balls of radius half the diameter, centered at the points of S, intersect pairwise and therefore by hyperconvexity have a common intersection; a ball of radius half the diameter centered at a point of this common intersection contains all of S. Thus, injective spaces satisfy a particularly strong form of Jung's theorem.
Every injective space is a complete space (Aronszajn and Panitchpakdi 1956), and every metric map (or, equivalently, nonexpansive mapping, or short map) on a bounded injective space has a fixed point (Sine 1979; Soardi 1979). A metric space is injective if and only if it is an injective object in the category of metric spaces and metric maps. For additional properties of injective spaces see Espínola and Khamsi (2001).
Read more about this topic: Injective Metric Space
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