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. 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.