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 that these two different types of definitions are equivalent.

Hyperconvexity

is said to be hyperconvex if it is convex and its closed balls have the binary Helly property. That is:

1. can be connected by the isometric is a path space).

2. .

).

Injectivity

A retraction is a function to a subspace of itself, such that

1. is the identity function on its image (i.e. it is idempotent), and

2. is nonexpansive.

A retract is said to be injective is isometric.

Examples

Examples of hyperconvex metric spaces include

• The real line

• with the \ell∞ distance

• Manhattan distance (L1) in the plane (which is equivalent up to rotation and scaling to the L∞), but not in higher dimensions

• The tight span of a metric space

• Any complete real tree

• see Metric space aimed at its subspace

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 is equal to half the diameter. Thus, injective spaces satisfy a particularly strong form of Jung's theorem.

Every injective space is a complete space, and every metric map (or, equivalently, nonexpansive mapping, or short map) on a bounded injective space has a fixed point. A metric space is injective if and only if it is an injective object in the category of metric spaces and metric maps.

Notes

References

• Correction (1957), Pacific J. Math. 7: 1729, .