Metric (mathematics)

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An illustration comparing the taxicab metric to the Euclidean metric on the plane: According to the taxicab metric the red, yellow, and blue paths have the same length (12). According to the Euclidean metric, the green path has length
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An illustration comparing the taxicab metric to the Euclidean metric on the plane: According to the taxicab metric the red, yellow, and blue paths have the same length (12). According to the Euclidean metric, the green path has length , and is the unique shortest path.

In mathematics, a metric or distance function is a function that gives a distance between each pair of point elements of a set. A set with a metric is called a metric space. [1] A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable.

Contents

One important source of metrics in differential geometry are metric tensors, bilinear forms that may be defined from the tangent vectors of a differentiable manifold onto a scalar. A metric tensor allows distances along curves to be determined through integration, and thus determines a metric.

Definition

A metric on a set X is a function (called distance function or simply distance)

where is the set of non-negative real numbers and for all , the following three axioms are satisfied:

1. identity of indiscernibles
2. symmetry
3. triangle inequality

A metric (as defined) is a non-negative real-valued function. This, together with axiom 1, provides a separation condition, where distinct or separate points are precisely those that have a positive distance between them.

The requirement that have a range of is a clarifying (but unnecessary) restriction in the definition, for if we had any function that satisfied the same three axioms, the function could be proven to still be non-negative as follows (using axioms 1, 3, and 2 in that order):

which implies .

A metric is called an ultrametric if it satisfies the following stronger version of the triangle inequality where points can never fall 'between' other points:

for all

A metric d on X is called intrinsic if any two points x and y in X can be joined by a curve with length arbitrarily close to d(x, y).

A metric d on a group G (written multiplicatively) is said to be left-invariant (resp. right invariant) if we have

[resp. ]

for all x, y, and z in G.

A metric on a commutative additive group is said to be translation invariant if for all or equivalently, if for all Every vector space is also a commutative additive group and a metric on a real or complex vector space that is induced by a norm is always translation invariant. A metric on a real or complex vector space is induced by a norm if and only if it is translation invariant and absolutely homogeneous, where the latter means that for all scalars and all in which case the function defines a norm on and the canonical metric induced by is equal to

Notes

These conditions express intuitive notions about the concept of distance. For example, that the distance between distinct points is positive and the distance from x to y is the same as the distance from y to x. The triangle inequality means that the distance from x to z via y is at least as great as from x to z directly. Euclid in his work stated that the shortest distance between two points is a line; that was the triangle inequality for his geometry.

Examples

Equivalence of metrics

For a given set X, two metrics d1 and d2 are called topologically equivalent (uniformly equivalent) if the identity mapping

id: (X,d1) → (X,d2)

is a homeomorphism (uniform isomorphism).

For example, if is a metric, then and are metrics equivalent to

Norm induced metric

Norms on vector spaces are equivalent to certain metrics, namely homogeneous, translation-invariant ones. In other words, every norm determines a metric, and some metrics determine a norm.

Given a normed vector space we can define a metric on called the metric induced by or simply the norm induced metric, by

The metric is said to be induced by the norm

Conversely [3] if a metric on a vector space satisfies the properties

then a norm on may be defined by

where the metric induced by this norm is the original given metric

Similarly, a seminorm induces a pseudometric (see below), and a homogeneous, translation invariant pseudometric induces a seminorm.

Metrics on multisets

We can generalize the notion of a metric from a distance between two elements to a distance between two nonempty finite multisets of elements. A multiset is a generalization of the notion of a set such that an element can occur more than once. Define if is the multiset consisting of the elements of the multisets and , that is, if occurs once in and once in then it occurs twice in . A distance function on the set of nonempty finite multisets is a metric [4] if

  1. if all elements of are equal and otherwise (positive definiteness), that is, (non-negativity plus identity of indiscernibles)
  2. is invariant under all permutations of (symmetry)
  3. (triangle inequality)

Note that the familiar metric between two elements results if the multiset has two elements in 1 and 2 and the multisets have one element each in 3. For instance if consists of two occurrences of , then according to 1.

A simple example is the set of all nonempty finite multisets of integers with . More complex examples are information distance in multisets; [4] and normalized compression distance (NCD) in multisets. [5]

Generalized metrics

There are numerous ways of relaxing the axioms of metrics, giving rise to various notions of generalized metric spaces. These generalizations can also be combined. The terminology used to describe them is not completely standardized. Most notably, in functional analysis pseudometrics often come from seminorms on vector spaces, and so it is natural to call them "semimetrics". This conflicts with the use of the term in topology.

Extended metrics

Some authors allow the distance function d to attain the value ∞, i.e. distances are non-negative numbers on the extended real number line. Such a function is called an extended metric or "∞-metric". Every extended metric can be transformed to a finite metric such that the metric spaces are equivalent as far as notions of topology (such as continuity or convergence) are concerned. This can be done using a subadditive monotonically increasing bounded function which is zero at zero, e.g. d(x, y) = d(x, y) / (1 + d(x, y)) or d(x, y) = min(1, d(x, y)).

The requirement that the metric take values in [0,∞) can even be relaxed to consider metrics with values in other directed sets. The reformulation of the axioms in this case leads to the construction of uniform spaces: topological spaces with an abstract structure enabling one to compare the local topologies of different points.

Pseudometrics

A pseudometric on X is a function which satisfies the axioms for a metric, except that instead of the second (identity of indiscernibles) only d(x,x) = 0 for all x is required. In other words, the axioms for a pseudometric are:

  1. d(x, y) ≥ 0
  2. d(x, x) = 0 (but possibly d(x, y) = 0 for some distinct values xy.)
  3. d(x, y) = d(y, x)
  4. d(x, z) ≤ d(x, y) + d(y, z).

In some contexts, pseudometrics are referred to as semimetrics because of their relation to seminorms.

Quasimetrics

Occasionally, a quasimetric is defined as a function that satisfies all axioms for a metric with the possible exception of symmetry. [6] The name of this generalisation is not entirely standardized. [7]

  1. d(x, y) ≥ 0 (positivity)
  2. d(x, y) = 0   if and only if  x = y (positive definiteness)
  3. d(x, y) = d(y, x) (symmetry, dropped)
  4. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

Quasimetrics are common in real life. For example, given a set X of mountain villages, the typical walking times between elements of X form a quasimetric because travel up hill takes longer than travel down hill. Another example is a taxicab geometry topology having one-way streets, where a path from point A to point B comprises a different set of streets than a path from B to A.

A quasimetric on the reals can be defined by setting

d(x, y) = xy if xy, and
d(x, y) = 1 otherwise. The 1 may be replaced by infinity or by .

The topological space underlying this quasimetric space is the Sorgenfrey line. This space describes the process of filing down a metal stick: it is easy to reduce its size, but it is difficult or impossible to grow it.

If d is a quasimetric on X, a metric d' on X can be formed by taking

d'(x, y) = 1/2(d(x, y) + d(y, x)).

Metametrics

In a metametric, all the axioms of a metric are satisfied except that the distance between identical points is not necessarily zero. In other words, the axioms for a metametric are:

  1. d(x, y) ≥ 0
  2. d(x, y) = 0 implies x = y (but not vice versa.)
  3. d(x, y) = d(y, x)
  4. d(x, z) ≤ d(x, y) + d(y, z).

Metametrics appear in the study of Gromov hyperbolic metric spaces and their boundaries. The visual metametric on such a space satisfies d(x, x) = 0 for points x on the boundary, but otherwise d(x, x) is approximately the distance from x to the boundary. Metametrics were first defined by Jussi Väisälä. [8]

Semimetrics

A semimetric on X is a function that satisfies the first three axioms, but not necessarily the triangle inequality:

  1. d(x, y) ≥ 0
  2. d(x, y) = 0   if and only if  x = y
  3. d(x, y) = d(y, x)

Some authors work with a weaker form of the triangle inequality, such as:

d(x, z) ≤ ρ (d(x, y) + d(y, z))   (ρ-relaxed triangle inequality)
d(x, z) ≤ ρ max(d(x, y), d(y, z))   (ρ-inframetric inequality).

The ρ-inframetric inequality implies the ρ-relaxed triangle inequality (assuming the first axiom), and the ρ-relaxed triangle inequality implies the 2ρ-inframetric inequality. Semimetrics satisfying these equivalent conditions have sometimes been referred to as "quasimetrics", [9] "nearmetrics" [10] or inframetrics. [11]

The ρ-inframetric inequalities were introduced to model round-trip delay times in the internet. [11] The triangle inequality implies the 2-inframetric inequality, and the ultrametric inequality is exactly the 1-inframetric inequality.

Premetrics

Relaxing the last three axioms leads to the notion of a premetric, i.e. a function satisfying the following conditions:

  1. d(x, y) ≥ 0
  2. d(x, x) = 0

This is not a standard term. Sometimes it is used to refer to other generalizations of metrics such as pseudosemimetrics [12] or pseudometrics; [13] in translations of Russian books it sometimes appears as "prametric". [14] It is also called a distance. [15]

Any premetric gives rise to a topology as follows. For a positive real r, the r-ball centered at a point p is defined as

Br(p) = { x | d(x, p) < r }.

A set is called open if for any point p in the set there is an r-ball centered at p which is contained in the set. Every premetric space is a topological space, and in fact a sequential space. In general, the r-balls themselves need not be open sets with respect to this topology. As for metrics, the distance between two sets A and B, is defined as

d(A, B) = infxA, yBd(x, y).

This defines a premetric on the power set of a premetric space. If we start with a (pseudosemi-)metric space, we get a pseudosemimetric, i.e. a symmetric premetric. Any premetric gives rise to a preclosure operator cl as follows:

cl(A) = { x | d(x, A) = 0 }.

Pseudoquasimetrics

The prefixes pseudo-, quasi- and semi- can also be combined, e.g., a pseudoquasimetric (sometimes called hemimetric) relaxes both the indiscernibility axiom and the symmetry axiom and is simply a premetric satisfying the triangle inequality. For pseudoquasimetric spaces the open r-balls form a basis of open sets. A very basic example of a pseudoquasimetric space is the set {0,1} with the premetric given by d(0,1) = 1 and d(1,0) = 0. The associated topological space is the Sierpiński space.

Sets equipped with an extended pseudoquasimetric were studied by William Lawvere as "generalized metric spaces". [16] From a categorical point of view, the extended pseudometric spaces and the extended pseudoquasimetric spaces, along with their corresponding nonexpansive maps, are the best behaved of the metric space categories. One can take arbitrary products and coproducts and form quotient objects within the given category. If one drops "extended", one can only take finite products and coproducts. If one drops "pseudo", one cannot take quotients. Approach spaces are a generalization of metric spaces that maintains these good categorical properties.

Łukaszyk-Karmowski distance

Łukaszyk-Karmowski distance is a function defining a distance between two random variables or two random vectors. The axioms of this function are:

  1. d(x, y) > 0
  2. d(x, y) = d(y, x)
  3. d(x, z) ≤ d(x, y) + d(y, z).

This distance function satisfies the identity of indiscernibles condition if and only if both arguments are described by idealized Dirac delta density probability distribution functions.

Important cases of generalized metrics

In differential geometry, one considers a metric tensor, which can be thought of as an "infinitesimal" quadratic metric function. This is defined as a nondegenerate symmetric bilinear form on the tangent space of a manifold with an appropriate differentiability requirement. While these are not metric functions as defined in this article, they induce what is called a pseudo-semimetric function by integration of its square root along a path through the manifold. If one imposes the positive-definiteness requirement of an inner product on the metric tensor, this restricts to the case of a Riemannian manifold, and the path integration yields a metric.

In general relativity the related concept is a metric tensor (general relativity) which expresses the structure of a pseudo-Riemannian manifold. Though the term "metric" is used, the fundamental idea is different because there are non-zero null vectors in the tangent space of these manifolds, and vectors can have negative squared norms. This generalized view of "metrics", in which zero distance does not imply identity, has crept into some mathematical writing too: [17]

See also

Notes

  1. Čech 1969.
  2. Narici & Beckenstein 2011, pp. 47–51.
  3. Narici & Beckenstein 2011, pp. 47-66.
  4. 1 2 Vitányi 2011.
  5. Cohen & Vitányi 2012.
  6. Steen & Seebach (1995); Smyth (1987)
  7. Rolewicz (1987) calls them "semimetrics". That same term is also frequently used for two other generalizations of metrics.
  8. Väisälä 2005.
  9. Xia 2009.
  10. Xia 2008.
  11. 1 2 Fraigniaud, Lebhar & Viennot 2008.
  12. Buldygin & Kozachenko 2000.
  13. Helemskii 2006.
  14. Arkhangel'skii & Pontryagin (1990); Aldrovandi & Pereira (2017)
  15. Deza & Laurent 1997.
  16. Lawvere (2002); Vickers (2005)
  17. Parrott (1987): "This bilinear form is variously called the Lorentz metric, or Minkowski metric or metric tensor"; Cecil (2008): "We call this scalar product the Lorentz metric"

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