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 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: Equivalently, a metric space $$X$$ is hyperconvex if, for any set of points $$p_i$$ in $$X$$ and radii $$r_i>0$$ satisfying $$r_i+r_j\ge d(p_i,p_j)$$ for each $$i$$ and $$j$$, there is a point $$q$$ in $$X$$ that is within distance $$r_i$$ of each $$p_i$$ (that is, $$d(p_i,q) \le r_i$$ for all $$i$$).
 * 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 (i.e. $$X$$ is a path space).
 * 2) If $$F$$ is any family of closed balls $${\bar B}_r(p) = \{q \mid d(p,q) \le r\}$$ such that each pair of balls in $$F$$ meets, then there exists a point $$x$$ common to all the balls in $$F$$.

Injectivity
A retraction of a metric space $$X$$ is a function $$f$$ 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 $$Y$$.
 * 1) 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
 * 2) for all $$x, y \in X$$ we have that $$d(f(x),f(y))\le d(x,y)$$; that is, $$f$$ is nonexpansive.

Examples
Examples of hyperconvex metric spaces include Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.
 * The real line
 * $$\R^d$$ 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

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, 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.