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.
Hyperconvexity A metric space X {\displaystyle 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 {\displaystyle x} and y {\displaystyle y} can be connected by the isometric image of a line segment of length equal to the distance between the points (i.e. X {\displaystyle X} is a path space). If F {\displaystyle F} is any family of closed balls B ¯ r ( p ) = { q ∣ d ( p , q ) ≤ r } {\displaystyle {\bar {B}}_{r}(p)=\{q\mid d(p,q)\leq r\}} such that each pair of balls in F {\displaystyle F} meets, then there exists a point x {\displaystyle x} common to all the balls in F {\displaystyle F} . Equivalently, a metric space X {\displaystyle X} is hyperconvex if, for any set of points p i {\displaystyle p_{i}} in X {\displaystyle X} and radii r i > 0 {\displaystyle r_{i}>0} satisfying r i + r j ≥ d ( p i , p j ) {\displaystyle r_{i}+r_{j}\geq d(p_{i},p_{j})} for each i {\displaystyle i} and j {\displaystyle j} , there is a point q {\displaystyle q} in X {\displaystyle X} that is within distance r i {\displaystyle r_{i}} of each p i {\displaystyle p_{i}} (that is, d ( p i , q ) ≤ r i {\displaystyle d(p_{i},q)\leq r_{i}} for all i {\displaystyle i} ).
Injectivity
A retraction of a metric space X {\displaystyle X} is a function f {\displaystyle f} mapping X {\displaystyle X} to a subspace of itself, such that
for all x ∈ X {\displaystyle x\in X} we have that f ( f ( x ) ) = f ( x ) {\displaystyle f(f(x))=f(x)} ; that is, f {\displaystyle f} is the identity function on its image (i.e. it is idempotent), and for all x , y ∈ X {\displaystyle x,y\in X} we have that d ( f ( x ) , f ( y ) ) ≤ d ( x , y ) {\displaystyle d(f(x),f(y))\leq d(x,y)} ; that is, f {\displaystyle f} is nonexpansive. A retract of a space X {\displaystyle X} is a subspace of X {\displaystyle X} that is an image of a retraction. A metric space X {\displaystyle X} is said to be injective if, whenever X {\displaystyle X} is isometric to a subspace Z {\displaystyle Z} of a space Y {\displaystyle Y} , that subspace Z {\displaystyle Z} is a retract of Y {\displaystyle Y} .
Examples Examples of hyperconvex metric spaces include
The real line
R d {\displaystyle \mathbb {R} ^{d}} with the ℓ {\displaystyle \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
Aim ( X ) {\displaystyle \operatorname {Aim} (X)} – see Metric space aimed at its subspace Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.
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