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 Lā 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] A metric space
is said to be hyperconvex if it is convex and its closed balls have the binary Helly property.
That is: Equivalently, a metric space
is hyperconvex if, for any set of points
A retraction of a metric space
to a subspace of itself, such that A retract of a space
that is an image of a retraction.
A metric space
is isometric to a subspace
Examples of hyperconvex metric spaces include Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.
In an injective space, the radius of the minimum ball that contains any set
is equal to half the diameter of
This follows since the balls of radius half the diameter, centered at the points of
, 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
Thus, injective spaces satisfy a particularly strong form of Jung's theorem.
Every injective space is a complete space,[2] and every metric map (or, equivalently, nonexpansive mapping, or short map) on a bounded injective space has a fixed point.
[3] A metric space is injective if and only if it is an injective object in the category of metric spaces and metric maps.