Injective metric space
Encyclopedia
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 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.
have the binary Helly property
. That is,
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.
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.
Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.
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 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 and Khamsi (2001).
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
with certain properties generalizing those of the real line and of L∞ distances
Chebyshev distance
In mathematics, Chebyshev distance , Maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension...
in higher-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
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
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
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 ballsBall (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
have the binary Helly property
Helly family
In combinatorics, a Helly family of order k is a family of sets such that any minimal subfamily with an empty intersection has k or fewer sets in it. In other words, any subfamily such that every k-fold intersection is non-empty has non-empty total intersection.The k-Helly property is the property...
. That is,
- any two points x and y can be connected by the isometric imageIsometryIn mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
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 ƒ mapping X to a subspace of itself, such that- for all x, ƒ(ƒ(x)) = ƒ(x); that is, ƒ is the identity functionIdentity functionIn mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
on its image, and - 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
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
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∞ distanceLp spaceIn mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
- Manhattan distanceTaxicab geometryTaxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their coordinates...
(L1) in the plane (which is equivalent up to rotation and scaling to the L∞), but not in higher dimensions - The tight spanTight spanIn metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes...
of a metric space - Any real treeReal treeA real tree, or an \mathbb R-tree, is a metric space such thatfor any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval [a,b] to M such that f=x and f=y...
- Aim(X) – see Metric space aimed at its subspaceMetric space aimed at its subspaceIn mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope, or tight span, which are basic objects of the category of metric spaces.Following , a notion of a metric...
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 diameterDiameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
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
Jung's theorem
In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set...
.
Every injective space is a complete space
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
(Aronszajn and Panitchpakdi 1956), and every metric map (or, equivalently, nonexpansive mapping, or short map
Short map
In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance .These maps are the morphisms in the category of metric spaces, Met ....
) on a bounded injective space has a fixed point
Fixed-point theorem
In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point , under some conditions on F that can be stated in general terms...
(Sine 1979; Soardi 1979). A metric space is injective if and only if it is an injective object
Injective object
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories...
in the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of metric spaces and metric maps
Category of metric spaces
In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map...
. For additional properties of injective spaces see Espínola and Khamsi (2001).