Metric (mathematics)

An illustration comparing the taxicab metric versus the Euclidean metric on the plane: In the taxicab metric all three pictured paths (red, yellow, and blue) have the same length (12) for the same route. In the Euclidean metric, the green path has length $6 \sqrt{2} \approx 8.49$, and is the unique shortest path.

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. A set with a metric is called a metric space. 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.

An 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. However, not every metric comes from a metric tensor in this way.

Definition

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

d : X × X → [0,∞),

where [0,∞) is the set of non-negative real numbers (because distance can't be negative so we can't use R), and for all x, y, z in X, the following conditions are satisfied:

1.	$d(x,y) \ge 0$	non-negativity or separation axiom
2.	$d(x,y) = 0 \Leftrightarrow x = y$	identity of indiscernibles
3.	$d(x,y) = d(y,x)$	symmetry
4.	$d(x,z) \le d(x,y) + d(y, z)$	subadditivity or triangle inequality

Conditions 1 and 2 together define a positive-definite function. The first condition is implied by the others.

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:

d(x, z) ≤ max(d(x, y), d(y, z))

for all x, y, z in X.

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).

For sets on which an addition + : X × X → X is defined, d is called a translation invariant metric if

d(x, y) = d(x + a, y + a)

for all x, y and a in X.

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.

If a modification of the triangle inequality

4*. d(x, z) ≤ d(z, y) + d(y, x)

is used in the definition then property 1 follows straight from property 4*. Properties 2 and 4* give property 3 which in turn gives property 4.

Examples

• The discrete metric: if x = y then d(x,y) = 0. Otherwise, d(x,y) = 1.
• The Euclidean metric is translation and rotation invariant.
• The taxicab metric is translation invariant.
• More generally, any metric induced by a norm is translation invariant.
• If $(p_n)_{n\in N}$ is a sequence of seminorms defining a (locally convex) topological vector space E, then

$d(x,y)=\sum_{n=1}^\infty \frac{1}{2^n} \frac{p_n(x-y)}{1+p_n(x-y)}$

is a metric defining the same topology. (One can replace $\frac{1}{2^n}$ by any summable sequence $(a_n)$ of strictly positive numbers.)

• Graph metric, a metric defined in terms of distances in a certain graph.
• The Hamming distance in coding theory.
• Riemannian metric, a type of metric function that is appropriate to impose on any differentiable manifold. For any such manifold, one chooses at each point p a symmetric, positive definite, bilinear form L: T_p × T_p → ℝ on the tangent space T_p at p, doing so in a smooth manner. This form determines the length of any tangent vector v on the manifold, via the definition ||v|| = √L(v, v). Then for any differentiable path on the manifold, its length is defined as the integral of the length of the tangent vector to the path at any point, where the integration is done with respect to the path parameter. Finally, to get a metric defined on any pair {x, y} of points of the manifold, one takes the infimum, over all paths from x to y, of the set of path lengths. A smooth manifold equipped with a Riemannian metric is called a Riemannian manifold.
• The Fubini–Study metric on complex projective space. This is an example of a Riemannian metric.
• String metrics, such as Levenshtein distance and other string edit distances, define a metric over strings.
• Graph edit distance defines a distance function between graphs.

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 $d$ is a metric, then $\min (d, 1)$ and ${d \over 1+d}$ are metrics equivalent to $d.$

See also notions of metric space equivalence.

Metrics on vector spaces

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 $(X, \|\cdot\|)$ we can define a metric on X by

$d(x,y) := \| x-y\|$.

The metric d is said to be induced by the norm $\|\cdot\|$.

Conversely if a metric d on a vector space X satisfies the properties

• $d(x,y) = d(x+a,y+a)$ (translation invariance)
• $d(\alpha x, \alpha y) = |\alpha| d(x,y)$ (homogeneity)

then we can define a norm on X by

$\|x\| := d(x,0)$

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 $Z=XY$ if $Z$ is the multiset consisting of the elements of the multisets $X$ and $Y$, that is, if $x$ occurs once in $X$ and once in $Y$ then it occurs twice in $Z$. A distance function $d$ on the set of nonempty finite multisets is a metric^[1] if

1. $d(X)=0$ if all elements of $X$ are equal and $d(X) > 0$ otherwise (positive definiteness), that is, (non-negativity plus identity of indiscernibles)
2. $d(X)$ is invariant under all permutations of $X$ (symmetry)
3. $d(XY) \leq d(XZ)+d(ZY)$ (triangle inequality)

Note that the familiar metric between two elements results if the multiset $X$ has two elements in 1 and 2 and the multisets $X,Y,Z$ have one element each in 3. For instance if $X$ consists of two occurrences of $x$, then $d(X)=0$ according to 1.

A simple example is the set of all nonempty finite multisets $X$ of integers with $d(X)=\max\{x: x \in X\}- \min\{x:x \in X\}$. More complex examples are information distance in multisets;^[1] and normalized compression distance (NCD) in multisets.^[2]

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 d : X × X → R 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 $x\ne y$.)
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:^[3][4]

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. Nevertheless, this notion is rarely used in mathematics, and its name is not entirely standardized.^[5]

A quasimetric on the reals can be defined by setting

d(x, y) = x − y if x ≥ y, and

d(x, y) = 1 otherwise. The 1 may be replaced by infinity or by $1+10^{(y-x)}$.

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.^[6]

Semimetrics

A semimetric on X is a function d : X × X → R 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",^[7] "nearmetrics"^[8] or inframetrics.^[9]

The ρ-inframetric inequalities were introduced to model round-trip delay times in the internet.^[9] 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^[10] or pseudometrics;^[11] in translations of Russian books it sometimes appears as "prametric".^[12]

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

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

A set is called open if for any point p in the set there is an "open" r-ball centred at p which is contained in the set. Every premetric space is a topological space, and in fact a sequential space. In general, the "open" 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) = inf_{x∊A, y∊B} d(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".^[13][14] 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.

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 in cosmology, the fundamental idea is different because there are non-zero null vectors in the tangent space of these manifolds. This generalized view of "metrics", in which zero distance does not imply identity, has crept into some mathematical writing too:^[15][16]

See also

• Acoustic metric
• Complete metric
• Similarity measure
• Signed distance function

Notes

1. P.M.B. Vitanyi, Information distance in multiples, IEEE Trans. Inform. Theory, 57:4(2011), 2451-2456
2. A.R.Cohen and P.M.B. Vitanyi, Normalized compression distance of multisets with applications, arXiv:1212.5711
3. E.g. Steen & Seebach (1995).
4. Smyth, M. (1987). M.Main, A.Melton, M.Mislove and D.Schmidt, ed. Quasi uniformities: reconciling domains with metric spaces. 3rd Conference on Mathematical Foundations of Programming Language Semantics. Springer-Verlag, Lecture Notes in Computer Science 298. pp. 236–253.
5. Rolewicz, Stefan (1987), Functional Analysis and Control Theory: Linear Systems, Springer, ISBN 90-277-2186-6, OCLC 13064804 This book calls them "semimetrics". That same term is also frequently used for two other generalizations of metrics.
6. Väisälä, Jussi (2005), "Gromov hyperbolic spaces" (PDF), Expositiones Mathematicae 23 (3): 187–231, doi:10.1016/j.exmath.2005.01.010, MR 2164775
7. Xia, Q. (2009), "The Geodesic Problem in Quasimetric Spaces", Journal of Geometric Analysis 19 (2): 452–479, doi:10.1007/s12220-008-9065-4
8. Qinglan Xia (2008), "The geodesic problem in nearmetric spaces", Journal of Geometric Analysis: Volume , Issue (009), Page 19 (2): 452–479, arXiv:0807.3377.
9. • Fraigniaud, P.; Lebhar, E.; Viennot, L. (2008), "2008 IEEE INFOCOM - The 27th Conference on Computer Communications", IEEE INFOCOM 2008. the 27th Conference on Computer Communications: 1085–1093, doi:10.1109/INFOCOM.2008.163, ISBN 978-1-4244-2026-1, retrieved 2009-04-17 |chapter= ignored (help).
10. Buldygin, V.V.; Kozachenko, I.U.V. (2000), Metric characterization of random variables and random processes.
11. Khelemskiĭ (2006), Lectures and exercises on functional analysis.
12. Arkhangel'skii & Pontryagin (1990). Aldrovandi, R.; Pereira, J.G. (1995), An introduction to geometrical physics.
13. Lawvere, F.W. (2002) [1973], Metric spaces, generalised logic, and closed categories, Reprints in Theory and Applications of Categories 1, pp. 1–37.
14. Vickers, Steven (2005), "Localic completion of generalized metric spaces I", Theory and Applications of Categories 14: 328–356
15. S. Parrott (1987) Relativistic Electrodynamics and Differential Geometry, page 4, Springer-Verlag ISBN 0-387-96435-5 : "This bilinear form is variously called the Lorentz metric, or Minkowski metric or metric tensor."
16. Thomas E. Cecil (1992) Lie Sphere Geometry, page 9, Springer-Verlag ISBN 0-387-97747-3 : "We call this scalar product the Lorentz metric"

References

• Arkhangel'skii, A. V.; Pontryagin, L. S. (1990), General Topology I: Basic Concepts and Constructions Dimension Theory, Encyclopaedia of Mathematical Sciences, Springer, ISBN 3-540-18178-4
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology, Dover, ISBN 978-0-486-68735-3, MR 507446, OCLC 32311847

External links

• Quasimetric space at PlanetMath.org.
• Semimetric at PlanetMath.org.