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.

Contents 1 Hyperconvexity 2 Injectivity 3 Examples 4 Properties 5 References

[edit] Hyperconvexity

A metric space is said to be hyperconvex if it is convex and its closed balls have the Helly property. That is,

1. 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
2. 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 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.

[edit] Injectivity

A retraction of a metric space X is a function f mapping X to a subspace of itself, such that

1. for all x, f(f(x)) = f(x); that is, f is the identity function on its image, and
2. 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 R^d with the L_∞ distance
• The tight span of a metric space
• Any real tree

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 on a bounded injective space has a fixed point (Sine 1979; Soardi 1979). 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 T_X approach to some results on cuts and metrics". Advances in Applied Mathematics 19 (4): 453–470. DOI:10.1006/aama.1997.0549. 

• 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. 

• Sine, R. C. (1979). "On linear contraction semigroups in sup norm spaces". Nonlinear Analysis 3: 885–890. 

• Soardi, P. (1979). "Existence of fixed points for nonexpansive mappings in certain Banach lattices". Proceedings of the American Mathematical Society 73: 25–29.