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 X is said to be hyperconvex if it is convex and its closed balls have the binary 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 (i.e. X is a path space), and
2. if 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 F meet, then there exists a point x common to all the balls in F.

Equivalently, if a set of points p_i and radii r_i > 0 satisfies r_i + r_j ≥ d(p_i, p_j) for each i and j, then there is a point q of the metric space that is within distance r_i of each p_i.

Injectivity[]

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

1. for all x, ƒ(ƒ(x)) = ƒ(x); that is, ƒ is the identity function on its image (i.e. it is idempotent), and
2. for all x and y, d(ƒ(x), ƒ(y)) ≤ d(x, y); that is, ƒ 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 (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) – 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 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 & Panitchpakdi 1956), and every metric map (or, equivalently, 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 metric maps. For additional properties of injective spaces see Espínola & Khamsi (2001).

References[]

• Aronszajn, N.; Panitchpakdi, P. (1956). "Extensions of uniformly continuous transformations and hyperconvex metric spaces". Pacific Journal of Mathematics. 6: 405–439. doi:10.2140/pjm.1956.6.405. MR 0084762. Correction (1957), Pacific J. Math. 7: 1729, MR0092146.
• 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. 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.
• Isbell, J. R. (1964). "Six theorems about injective metric spaces". Commentarii Mathematici Helvetici. 39: 65–76. doi:10.1007/BF02566944. MR 0182949.
• Sine, R. C. (1979). "On nonlinear contraction semigroups in sup norm spaces". Nonlinear Analysis. 3 (6): 885–890. doi:10.1016/0362-546X(79)90055-5. MR 0548959.
• Soardi, P. (1979). "Existence of fixed points of nonexpansive mappings in certain Banach lattices". Proceedings of the American Mathematical Society. 73 (1): 25–29. doi:10.2307/2042874. JSTOR 2042874. MR 0512051.