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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]

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).
If $F$ is any family of closed balls ${\bar {B}}_{r}(p)=\{q\mid d(p,q)\leq r\}$ such that each pair of balls in $F$ meets, then there exists a point $x$ common to all the balls in $F$ .

Equivalently, a metric space $X$ is hyperconvex if, for any set of points $p_{i}$ in $X$ and radii $r_{i}>0$ satisfying $r_{i}+r_{j}\geq d(p_{i},p_{j})$ for each $i$ and $j$ , there is a point $q$ in $X$ that is within distance $r_{i}$ of each $p_{i}$ (that is, $d(p_{i},q)\leq r_{i}$ for all $i$ ).

Injectivity

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

for all $x\in X$ we have that $f(f(x))=f(x)$ ; that is, $f$ is the identity function on its image (i.e. it is idempotent), and
for all $x,y\in X$ we have that $d(f(x),f(y))\leq d(x,y)$ ; that is, $f$ 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
$\mathbb {R} ^{d}$ 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
$\operatorname {Aim} (X)$ – 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.