Injective metric space

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 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:::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 "pi" and positive radii "ri" has the property that, for each "i" and "j", "ri" + "rj" ≥ "d"("pi","pj"), then there is a point "q" of the metric space that is within distance "ri" of each "pi".

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 the identity 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 Rd with the L 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 real tree
* Aim(X) &ndash; see Metric space aimed at its subspaceDue 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 of Jung'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 an injective 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 "TX" 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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