# 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.

## Hyperconvexity

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

- 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 (i.e.*X*is a path space), and - if
*F*is any family of closed balls

- such that each pair of balls in
*F*meet, then there exists a point*x*common to all the balls in*F*.

Equivalently, if a set of points *p _{i}* and radii

*r*satisfies

_{i}> 0*r*+

_{i}*r*≥

_{j}*d*(

*p*,

_{i}*p*) for each

_{j}*i*and

*j*, then there is a point

*q*of the metric space that is within distance

*r*of each

_{i}*p*.

_{i}## Injectivity

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

- for all
*x*,*ƒ*(*ƒ*(*x*)) =*ƒ*(*x*); that is,*ƒ*is the identity function on its image (i. e. it is idempotent), and - 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

- The real line
- Any vector space
**R**^{d}with the L_{∞}distance - Manhattan distance (
*L*_{1}) 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 real tree
- Aim(
*X*) – 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 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 & 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 & Khamsi (2001).

## References

- Aronszajn, N.; Panitchpakdi, P. (1956). "Extensions of uniformly continuous transformations and hyperconvex metric spaces".
*Pacific Journal of Mathematics*.**6**: 405–439. doi:10.2140/pjm.1956.6.405. MR 0084762. Correction (1957),*Pacific J. Math.***7**: 1729, MR0092146. - Chepoi, Victor (1997). "A
*T*approach to some results on cuts and metrics"._{X}*Advances in Applied Mathematics*.**19**(4): 453–470. doi:10.1006/aama.1997.0549. MR 1479014. - Espínola, R.; Khamsi, M. A. (2001). "Introduction to hyperconvex spaces" (PDF). In Kirk, W. A.; Sims B. (eds.).
*Handbook of Metric Fixed Point Theory*. Dordrecht: Kluwer Academic Publishers. MR 1904284. - Isbell, J. R. (1964). "Six theorems about injective metric spaces".
*Commentarii Mathematici Helvetici*.**39**: 65–76. doi:10.1007/BF02566944. MR 0182949. - Sine, R. C. (1979). "On nonlinear contraction semigroups in sup norm spaces".
*Nonlinear Analysis*.**3**(6): 885–890. doi:10.1016/0362-546X(79)90055-5. MR 0548959. - Soardi, P. (1979). "Existence of fixed points of nonexpansive mappings in certain Banach lattices".
*Proceedings of the American Mathematical Society*.**73**(1): 25–29. doi:10.2307/2042874. JSTOR 2042874. MR 0512051.