Injective metric space
In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of Aronszajn and Panitchpakdi (1956; see e.g. Chepoi 1997) that these two different types of definitions are equivalent.
A metric space X is said to be hyperconvex if it is convex and its closed balls have the binary Helly property. That is,
Equivalently, if a set of points pi and radii ri > 0 satisfies ri + rj ≥ d(pi,pj) for each i and j, then there is a point q of the metric space that is within distance ri of each pi.
A retraction of a metric space X is a function ƒ mapping X to a subspace of itself, such that
A retract of a space X is a subspace of X that is an image of a retraction. A metric space X is said to be injective if, whenever X is isometric to a subspace Z of a space Y, that subspace Z is a retract of Y.
Examples of hyperconvex metric spaces include
Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.
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.
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