- Injective metric space
In

metric geometry , an**injective metric space**, or equivalently a**hyperconvex metric space**, is ametric space with certain properties generalizing those of the real line and of L_{∞}distances in higher-dimensionalvector space s. 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 and Panitchpakdi (1956; see e.g. Chepoi 1997) that these two different types of definitions are equivalent.**Hyperconvexity**A metric space 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, and

#if "F" is any family of sets of the form::$\{ar\; B\}\_r(p)\; =\; \{q\; mid\; d(p,q)\; le\; r\}$:and if all pairs of sets in "F" intersect, then there exists a point "x" belonging to all sets in "F".An equivalent definition is that, if a set of points "p

_{i}" and positive radii "r_{i}" has the property that, for each "i" and "j", "r_{i}" + "r_{j}" ≥ "d"("p_{i}","p_{j}"), then there is a point "q" of the metric space that is within distance "r_{i}" of each "p_{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", "f(f(x)) = f(x)"; that is, "f" is theidentity function on its image, and

# for all "x" and "y", "d(f(x),f(y)) ≤ 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

* Any vector space**R**^{d}with the L_{∞}distance

* Manhattan distance (L_{1}) in the plane (which is equivalent up to rotation and scaling to the L_{∞}), but not in higher dimensions

* Thetight span of a metric space

* Anyreal tree

* Aim(X) – seeMetric 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 ofJung's theorem .Every injective space is a

complete space (Aronszajn and Panitchpakdi 1956), and every nonexpansive mapping, or short map, on a bounded injective space has a fixed point (Sine 1979; Soardi 1979). A metric space is injective if and only if it is aninjective object in the category of metric spaces and non-expansive maps. For additional properties of injective spaces see Espínola and Khamsi (2001).**References***cite journal

author = Aronszajn, N.; Panitchpakdi, P.

title = Extensions of uniformly continuous transformations and hyperconvex metric spaces

journal = Pacific Journal of Mathematics

volume = 6

year = 1956

pages = 405–439

url = http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103043960*cite journal

author = Chepoi, Victor

title = A "T_{X}" approach to some results on cuts and metrics

journal = Advances in Applied Mathematics

volume = 19

issue = 4

year = 1997

pages = 453–470

doi = 10.1006/aama.1997.0549*cite conference

author = Espínola, R.; Khamsi, M. A.

title = Introduction to hyperconvex spaces

booktitle = Handbook of Metric Fixed Point Theory

editor = Kirk, W. A.; Sims, B. (Eds.)

publisher = Kluwer Academic Publishers

location = Dordrecht

year = 2001

url = http://drkhamsi.com/publication/Es-Kh.pdf*cite journal

author = Isbell, J. R.

title = Six theorems about injective metric spaces

journal = Comment. Math. Helv.

volume = 39

year = 1964

pages = 65–76

doi = 10.1007/BF02566944*cite journal

author = Sine, R. C.

title = On linear contraction semigroups in sup norm spaces

journal = Nonlinear Analysis

volume = 3

year = 1979

pages = 885–890

doi = 10.1016/0362-546X(79)90055-5*cite journal

author = Soardi, P.

title = Existence of fixed points for nonexpansive mappings in certain Banach lattices

journal = Proceedings of the American Mathematical Society

volume = 73

year = 1979

pages = 25–29

doi = 10.2307/2042874

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