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.

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Hyperconvexity

A metric space is said to be hyperconvex if it is convex and its closed balls have the binary Helly property. That is,

  1. any two points x and y can be connected by the isometric image of a line segment of length equal to the distance between the points, and
  2. if F is any family of sets of the form
{\bar B}_r(p) = \{q \mid d(p,q) \le r\}
and if all pairs of sets in F intersect, then there exists a point x belonging to all sets in F.

An equivalent definition is that, if a set of points pi and positive radii ri has the property that, for each i and j, ri + rj ≥ d(pi,pj), then there is a point q of the metric space that is within distance ri of each pi.

Injectivity

A retraction of a metric space X is a function ƒ mapping X to a subspace of itself, such that

  1. for all x, ƒ(ƒ(x)) = ƒ(x); that is, ƒ is the identity function on its image, and
  2. for all x and y, d(ƒ(x), ƒ(y)) ≤ d(x, y); that is, ƒ is nonexpansive.

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

Examples of hyperconvex metric spaces include

Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.

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).

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