Injective metric space

In

Hyperconvexity

A metric space ${\displaystyle X}$ is said to be hyperconvex if it is

. That is:

1. Any two points ${\displaystyle x}$ and ${\displaystyle y}$ can be connected by the isometric image of a line segment of length equal to the distance between the points (i.e. ${\displaystyle X}$ is a path space).
2. If ${\displaystyle F}$ is any family of closed balls
   $${\displaystyle {\bar {B}}_{r}(p)=\{q\mid d(p,q)\leq r\}}$$

   
   such that each pair of balls in ${\displaystyle F}$ meets, then there exists a point ${\displaystyle x}$ common to all the balls in ${\displaystyle F}$.

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

Injectivity

A

of a metric space ${\displaystyle X}$ is a function ${\displaystyle f}$ mapping ${\displaystyle X}$ to a subspace of itself, such that

1. for all ${\displaystyle x\in X}$ we have that ${\displaystyle f(f(x))=f(x)}$; that is, ${\displaystyle f}$ is the
   idempotent
   ), and
2. for all ${\displaystyle x,y\in X}$ we have that ${\displaystyle d(f(x),f(y))\leq d(x,y)}$; that is, ${\displaystyle f}$ is
   nonexpansive
   .

A retract of a space ${\displaystyle X}$ is a subspace of ${\displaystyle X}$ that is an image of a retraction. A metric space ${\displaystyle X}$ is said to be injective if, whenever ${\displaystyle X}$ is isometric to a subspace ${\displaystyle Z}$ of a space ${\displaystyle Y}$, that subspace ${\displaystyle Z}$ is a retract of ${\displaystyle Y}$.

Examples

Examples of hyperconvex metric spaces include

• The real line
• ${\displaystyle \mathbb {R} ^{d}}$ with the ${\displaystyle \ell }$^∞ 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
• ${\displaystyle \operatorname {Aim} (X)}$ – see Metric space aimed at its subspace

Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.

Properties

In an injective space, the radius of the

that contains any set ${\displaystyle S}$ is equal to half the diameter of ${\displaystyle S}$. This follows since the balls of radius half the diameter, centered at the points of ${\displaystyle S}$, 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 ${\displaystyle S}$. Thus, injective spaces satisfy a particularly strong form of Jung's theorem.

Every injective space is a

Notes

References

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  .
• Chepoi, Victor (1997). "A T_X approach to some results on cuts and metrics".
  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
  .
• MR 0182949
  .
• Sine, R. C. (1979). "On nonlinear contraction semigroups in sup norm spaces". Nonlinear Analysis. 3 (6): 885–890.
  MR 0548959
  .
• Soardi, P. (1979). "Existence of fixed points of nonexpansive mappings in certain Banach lattices".
  MR 0512051
  .