HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the triangle inequality states that for any
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If , , and are the lengths of the sides of the triangle, with no side being greater than , then the triangle inequality states that :z \leq x + y , with equality only in the degenerate case of a triangle with zero area. In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths ( norms): :\, \mathbf x + \mathbf y\, \leq \, \mathbf x\, + \, \mathbf y\, , where the length of the third side has been replaced by the vector sum . When and are
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, they can be viewed as vectors in , and the triangle inequality expresses a relationship between
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
s. In Euclidean geometry, for
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
s the triangle inequality is a consequence of the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, and for general triangles, a consequence of the
law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
, although it may be proven without these theorems. The inequality can be viewed intuitively in either or . The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a angle and two angles, making the three vertices
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. In
spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
, the shortest distance between two points is an arc of a
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in ) with those endpoints. The triangle inequality is a ''defining property'' of norms and measures of
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s,
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s, the Lp spaces (), and
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
s.


Euclidean geometry

Euclid proved the triangle inequality for distances in
plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
using the construction in the figure. Beginning with triangle , an isosceles triangle is constructed with one side taken as and the other equal leg along the extension of side . It then is argued that angle has larger measure than angle , so side is longer than side . But , so the sum of the lengths of sides and is larger than the length of . This proof appears in
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...
, Book 1, Proposition 20.


Mathematical expression of the constraint on the sides of a triangle

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths , , that are all positive and excludes the degenerate case of zero area): :a + b > c ,\quad b + c > a ,\quad c + a > b . A more succinct form of this inequality system can be shown to be :, a - b, < c < a + b . Another way to state it is :\max(a, b, c) < a + b + c - \max(a, b, c) implying :2 \max(a, b, c) < a + b + c and thus that the longest side length is less than the
semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
. A mathematically equivalent formulation is that the area of a triangle with sides ''a'', ''b'', ''c'' must be a real number greater than zero.
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
for the area is : \begin 4\cdot \text & =\sqrt \\ & = \sqrt. \end In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero). The triangle inequality provides two more interesting constraints for triangles whose sides are ''a, b, c'', where ''a'' ≥ ''b'' ≥ ''c'' and \phi is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
, as :1<\frac<3 :1\le\min\left(\frac, \frac\right)<\phi.


Right triangle

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum. The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle . An isosceles triangle is constructed with equal sides . From the
triangle postulate A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...
, the angles in the right triangle satisfy: : \alpha + \gamma = \pi /2 \ . Likewise, in the isosceles triangle , the angles satisfy: :2\beta + \gamma = \pi \ . Therefore, : \alpha = \pi/2 - \gamma ,\ \mathrm \ \beta= \pi/2 - \gamma /2 \ , and so, in particular, :\alpha < \beta \ . That means side opposite angle is shorter than side opposite the larger angle . But . Hence: :\overline > \overline \ . A similar construction shows , establishing the theorem. An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point : (i) as depicted (which is to be proven), or (ii) coincident with (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle , which would violate the
triangle postulate A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...
), or lastly, (iii) interior to the right triangle between points and (in which case angle is an exterior angle of a right triangle and therefore larger than , meaning the other base angle of the isosceles triangle also is greater than and their sum exceeds in violation of the triangle postulate). This theorem establishing inequalities is sharpened by
Pythagoras' theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.


Examples of use

Consider a triangle whose sides are in an
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
and let the sides be , , . Then the triangle inequality requires that : 0 : 0 : 0 To satisfy all these inequalities requires : a>0 \text -\frac When is chosen such that , it generates a right triangle that is always similar to the
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
with sides , , . Now consider a triangle whose sides are in a
geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...
and let the sides be , , . Then the triangle inequality requires that : 0 : 0 : 0 The first inequality requires ; consequently it can be divided through and eliminated. With , the middle inequality only requires . This now leaves the first and third inequalities needing to satisfy : \begin r^2+r-1 & >0 \\ r^2-r-1 & <0. \end The first of these quadratic inequalities requires to range in the region beyond the value of the positive root of the quadratic equation , i.e. where is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
. The second quadratic inequality requires to range between 0 and the positive root of the quadratic equation , i.e. . The combined requirements result in being confined to the range :\varphi - 1 < r <\varphi\, \text a >0. When the common ratio is chosen such that it generates a right triangle that is always similar to the
Kepler triangle A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Squares ...
.


Generalization to any polygon

The triangle inequality can be extended by
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.


Example of the generalized polygon inequality for a quadrilateral

Consider a quadrilateral whose sides are in a
geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...
and let the sides be , , , . Then the generalized polygon inequality requires that : 0 : 0 : 0 : 0 These inequalities for reduce to the following : r^3+r^2+r-1>0 : r^3-r^2-r-1<0. The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, is limited to the range where is the tribonacci constant.


Relationship with shortest paths

This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line. No polygonal path between two points is shorter than the line between them. This implies that no curve can have an
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
less than the distance between its endpoints. By definition, the arc length of a curve is the
least upper bound In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path. p. 95.


Converse

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths. In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists. By the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
we have and according to the figure at the right. Subtracting these yields . This equation allows us to express in terms of the sides of the triangle: :d=\frac. For the height of the triangle we have that . By replacing with the formula given above, we have :h^2 = b^2-\left(\frac\right)^2. For a real number ''h'' to satisfy this, h^2 must be non-negative: :b^2-\left (\frac\right) ^2 \ge 0, :\left( b- \frac\right) \left( b+ \frac\right) \ge 0, :\left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \ge 0, :(a+b-c)(a-b+c)(b+c+a)(b+c-a) \ge 0, :(a+b-c)(a+c-b)(b+c-a) \ge 0, which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds
strictly In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive ...
, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.


Generalization to higher dimensions

The area of a triangular face of a
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an -
facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...
of an -
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
is less than or equal to the sum of the hypervolumes of the other facets. Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a
polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets. In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points , , , and in Euclidean space such that distances : and :. However, points with such distances cannot exist: the area of the 26–26–26 equilateral triangle is , which is larger than three times , the area of a 26–14–14 isosceles triangle (all by
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
), and so the arrangement is forbidden by the tetrahedral inequality.


Normed vector space

In a
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
, one of the defining properties of the
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
is the triangle inequality: : \, x + y\, \leq \, x\, + \, y\, \quad \forall \, x, y \in V that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as
subadditivity In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
. For any proposed function to behave as a norm, it must satisfy this requirement. If the normed space is euclidean, or, more generally, strictly convex, then \, x+y\, =\, x\, +\, y\, if and only if the triangle formed by , , and , is degenerate, that is, and are on the same ray, i.e., or , or for some . This property characterizes strictly convex normed spaces such as the spaces with . However, there are normed spaces in which this is not true. For instance, consider the plane with the norm (the
Manhattan distance A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences ...
) and denote and . Then the triangle formed by , , and , is non-degenerate but :\, x+y\, =\, (1,1)\, =, 1, +, 1, =2=\, x\, +\, y\, .


Example norms

*''Absolute value as norm for the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), 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 ...
.'' To be a norm, the triangle inequality requires that the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
satisfy for any real numbers and : , x + y, \leq , x, +, y, , which it does. Proof: :-\left\vert x \right\vert \leq x \leq \left\vert x \right\vert :-\left\vert y \right\vert \leq y \leq \left\vert y \right\vert After adding, :-( \left\vert x \right\vert + \left\vert y \right\vert ) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert Use the fact that \left\vert b \right\vert \leq a \Leftrightarrow -a \leq b \leq a (with ''b'' replaced by ''x''+''y'' and ''a'' by \left\vert x \right\vert + \left\vert y \right\vert), we have :, x + y, \leq , x, +, y, The triangle inequality is useful in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers. There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers and : :, x-y, \geq \biggl, , x, -, y, \biggr, . *''Inner product as norm in an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
as follows: Given vectors x and y, and denoting the inner product as \langle x , y\rangle : : The Cauchy–Schwarz inequality turns into an equality if and only if and are linearly dependent. The inequality \langle x, y \rangle + \langle y, x \rangle \le 2\left, \left\langle x, y \right\rangle\ turns into an equality for linearly dependent x and y if and only if one of the vectors or is a ''nonnegative'' scalar of the other. :Taking the square root of the final result gives the triangle inequality. * -norm: a commonly used norm is the ''p''-norm: \, x\, _p = \left( \sum_^n , x_i, ^p \right) ^ \ , where the are the components of vector . For the -norm becomes the ''Euclidean norm'': \, x\, _2 = \left( \sum_^n , x_i, ^2 \right) ^ = \left( \sum_^n x_^2 \right) ^ \ , which is
Pythagoras' theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
in -dimensions, a very special case corresponding to an inner product norm. Except for the case , the -norm is ''not'' an inner product norm, because it does not satisfy the
parallelogram law In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...
. The triangle inequality for general values of is called
Minkowski's inequality In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces. Let ''S'' be a measure space, let and let ''f'' and ''g'' be elements of L''p''(''S''). Then is in L''p''(''S''), and we have the tr ...
. It takes the form:\, x+y\, _p \le \, x\, _p + \, y\, _p \ .


Metric space

In 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 settin ...
with metric , the triangle inequality is a requirement upon
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
: :d(x,\ z) \le d(x,\ y) + d(y,\ z) \ , for all , , in . That is, the distance from to is at most as large as the sum of the distance from to and the distance from to . The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any
convergent sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...
in a metric space is a
Cauchy sequence In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
is a direct consequence of the triangle inequality, because if we choose any and such that and , where is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, , so that the sequence is a Cauchy sequence, by definition. This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via , with being the vector pointing from point to .


Reverse triangle inequality

The reverse triangle inequality is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is: :''Any side of a triangle is greater than or equal to the difference between the other two sides''. In the case of a normed vector space, the statement is: : \bigg, \, x\, -\, y\, \bigg, \leq \, x-y\, , or for metric spaces, . This implies that the norm \, \cdot\, as well as the distance function d(x,\cdot) are
Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
with Lipschitz constant , and therefore are in particular
uniformly continuous In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
. The proof for the reverse triangle uses the regular triangle inequality, and \, y-x\, = \, 1(x-y)\, = , 1, \cdot\, x-y\, = \, x-y\, : : \, x\, = \, (x-y) + y\, \leq \, x-y\, + \, y\, \Rightarrow \, x\, - \, y\, \leq \, x-y\, , : \, y\, = \, (y-x) + x\, \leq \, y-x\, + \, x\, \Rightarrow \, x\, - \, y\, \geq -\, x-y\, , Combining these two statements gives: : -\, x-y\, \leq \, x\, -\, y\, \leq \, x-y\, \Rightarrow \bigg, \, x\, -\, y\, \bigg, \leq \, x-y\, .


Triangle inequality for cosine similarity

By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that \operatorname(x,z) \geq \operatorname(x,y) \cdot \operatorname(y,z) - \sqrt and \operatorname(x,z) \leq \operatorname(x,y) \cdot \operatorname(y,z) + \sqrt\,. With these formulas, one needs to compute a
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
for each triple of vectors that is examined rather than for each pair of vectors examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.


Reversal in Minkowski space

The
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
metric \eta_ is not positive-definite, which means that \, x\, ^2 = \eta_ x^\mu x^\nu can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed: : \, x+y\, \geq \, x\, + \, y\, . A physical example of this inequality is the
twin paradox In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. Thi ...
in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.


See also

*
Subadditivity In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
*
Minkowski inequality In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces. Let ''S'' be a measure space, let and let ''f'' and ''g'' be elements of L''p''(''S''). Then is in L''p''(''S''), and we have the t ...
*
Ptolemy's inequality In Euclidean geometry, Ptolemy's inequality relates the six distances determined by four points in the plane or in a higher-dimensional space. It states that, for any four points , , , and , the following inequality holds: :\overline\cdot \overli ...


Notes


References

* . * .


External links

{{DEFAULTSORT:Triangle Inequality Geometric inequalities Linear algebra Metric geometry Articles containing proofs Theorems in geometry