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 & Panitchpakdi (1956) that these two different types of definitions are equivalent.[1]
A metric space is said to be hyperconvex if it is convex and its closed balls have the binary Helly property. That is:
Equivalently, a metric space is hyperconvex if, for any set of points in and radii satisfying for each and , there is a point in that is within distance of each (that is, for all ).
A retraction of a metric space is a function mapping to a subspace of itself, such that
A retract of a space is a subspace of that is an image of a retraction. A metric space is said to be injective if, whenever is isometric to a subspace of a space , that subspace is a retract of .
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 is equal to half the diameter of . This follows since the balls of radius half the diameter, centered at the points of , 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 . Thus, injective spaces satisfy a particularly strong form of Jung's theorem.
Every injective space is a complete space,[2] and every metric map (or, equivalently, nonexpansive mapping, or short map) on a bounded injective space has a fixed point.[3] A metric space is injective if and only if it is an injective object in the category of metric spaces and metric maps.[4]