In
metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
with certain properties generalizing those of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
and of
L∞ distances in higher-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 that these two different types of definitions are equivalent.
Hyperconvexity
A metric space
is said to be hyperconvex if it is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
and its
closed balls have the binary
Helly property
In combinatorics, a Helly family of order is a family of sets in which every minimal ''subfamily with an empty intersection'' has or fewer sets in it. Equivalently, every finite subfamily such that every -fold intersection is non-empty has non ...
. That is:
#Any two points
and
can be connected by the
isometric image of a line segment of length equal to the distance between the points (i.e.
is a path space).
#If
is any family of closed balls
such that each pair of balls in
meets, then there exists a point
common to all the balls in
. Equivalently, if a set of points
and
radii satisfies
for each
and
, then there is a point
of the metric space that is within distance
of each
.
Injectivity
A
retraction of a metric space
is a
function mapping
to a subspace of itself, such that
# for all
,
; that is,
is the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on its image (i.e. it is
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
), and
# for all
and
,
; that is,
is
nonexpansive.
A ''retract'' of a space
is a subspace of
that is an image of a retraction.
A metric space
is said to be injective if, whenever
is
isometric to a subspace
of a space
, that subspace
is a retract of
.
Examples
Examples of hyperconvex metric spaces include
* The real line
* Any vector space
with the
''L''∞ distance
*
Manhattan distance
A taxicab geometry or a Manhattan geometry is a geometry whose 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 Cartesian co ...
(''L''
1) in the plane (which is equivalent up to rotation and scaling to the ''L''
∞), but not in higher dimensions
* The
tight span
In 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 ...
of a metric space
* Any
real tree
*
– see
Metric space aimed at its subspace In 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 (injective) object ...
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
is equal to half the
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
of
. This follows since the balls of radius half the diameter, centered at the points of
, 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
. Thus, injective spaces satisfy a particularly strong form of
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. It is named after Heinrich Jung, who first studied this inequality in 1901. Alg ...
.
Every injective space is a
complete space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
, and every
metric map In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous).
These maps are the morphisms in the category of metric spaces, Met (Isbell 19 ...
(or, equivalently,
nonexpansive mapping, or short map) on a bounded injective space has a
fixed point. A metric space is injective
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
it is an
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 cohomology, in homotopy theory and in the theory of model categori ...
in the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of
metric spaces and metric maps.
[For additional properties of injective spaces see .]
Notes
References
* Correction (1957), ''Pacific J. Math.'' 7: 1729, .
*
*
*
*
*{{cite journal
, last = Soardi , first = P.
, title = Existence of fixed points of nonexpansive mappings in certain Banach lattices
, mr = 0512051
, journal =
Proceedings of the American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages.
According to the ' ...
, volume = 73
, year = 1979
, pages = 25–29
, doi = 10.2307/2042874
, issue = 1
, jstor = 2042874
, doi-access = free
Metric geometry