Injective metric space
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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.
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[edit] 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.
[edit] 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.
[edit] Examples
Examples of hyperconvex metric spaces include
- The real line
- Manhattan distance in the plane
- Any vector space Rd with the L∞ distance
- The tight span of a metric space
- Any real tree
- Aim(X) – see Metric space aimed at its subspace
Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.
[edit] 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).
[edit] References
- Aronszajn, N.; Panitchpakdi, P. (1956). "Extensions of uniformly continuous transformations and hyperconvex metric spaces". Pacific Journal of Mathematics 6: 405–439.
- Chepoi, Victor (1997). "A TX approach to some results on cuts and metrics". Advances in Applied Mathematics 19 (4): 453–470. doi:.
- Espínola, R.; Khamsi, M. A. (2001). "Introduction to hyperconvex spaces". Kirk, W. A.; Sims, B. (Eds.) Handbook of Metric Fixed Point Theory, Dordrecht: Kluwer Academic Publishers.
- Isbell, J. R. (1964). "Six theorems about injective metric spaces". Comment. Math. Helv. 39: 65–76. doi:.
- Sine, R. C. (1979). "On linear contraction semigroups in sup norm spaces". Nonlinear Analysis 3: 885–890. doi:.
- Soardi, P. (1979). "Existence of fixed points for nonexpansive mappings in certain Banach lattices". Proceedings of the American Mathematical Society 73: 25–29. doi:.
