TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the triangle inequality states that for any
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

, 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 , 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 Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a sma ...
and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (
norms Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative ethics that is prescriptive rather than a descriptive or explanato ...
): :$\, \mathbf x + \mathbf y\, \leq \, \mathbf x\, + \, \mathbf y\, ,$ where the length of the third side has been replaced by the vector sum . When and are
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, they can be viewed as vectors in , and the triangle inequality expresses a relationship between
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

s. In Euclidean geometry, for
right triangle A right triangle (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Currently, American Eng ...

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 among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

, and for general triangles, a consequence of the
law of cosines In trigonometry Trigonometry (from Greek '' trigōnon'', "triangle" and '' metron'', "measure") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathem ...
, 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. In
spherical geometry Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small tr ...
, the shortest distance between two points is an arc of a
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

, 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 Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative ethics that is prescriptive rather than a descriptive or explanato ...
and measures of
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

. 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s,
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
s, the Lp spaces (), and
inner product space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s.

# Euclidean geometry

Euclid proved the triangle inequality for distances in
plane geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a small ...
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 Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, ...
, 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 :$\max\left(a, b, c\right) < a + b + c - \max\left(a, b, c\right)$ implying :$2 \max\left(a, b, c\right) < a + b + c$ and thus that the longest side length is less than the
semiperimeter In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

. 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, as :$1<\frac<3$ :$1\le\min\left\left(\frac, \frac\right\right)<\phi.$

## 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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is conce ...
, 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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is conce ...
), 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...
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 (AP) 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 diffe ...

and let the sides be , , . Then the triangle inequality requires that :
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

and let the sides be , , . Then the triangle inequality requires that :
golden ratio In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

. 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 .

## Generalization to any polygon

The triangle inequality can be extended by
mathematical induction Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. Mathematical induction is a mathematical proof A mathematical proof is an Inference, inferential Argument-deduction-proof distinct ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

and let the sides be , , , . Then the generalized polygon inequality requires that :

### 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 length is the distance between two points along a section of a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. In ...

less than the distance between its endpoints. By definition, the arc length of a curve is the
least upper bound In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

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 :$h^2 = b^2-\left\left(\frac\right\right)^2.$ For a real number ''h'' to satisfy this, $h^2$ must be non-negative: :$b^2-\left \left(\frac\right\right) ^2 \ge 0,$ :$\left\left( b- \frac\right\right) \left\left( b+ \frac\right\right) \ge 0,$ :$\left\left(a^2-\left(b-c\right)^2\right)\left(\left(b+c\right)^2-a^2 \right\right) \ge 0,$ :$\left(a+b-c\right)\left(a-b+c\right)\left(b+c+a\right)\left(b+c-a\right) \ge 0,$ :$\left(a+b-c\right)\left(a+c-b\right)\left(b+c-a\right) \ge 0,$ 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 mathematics, mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of Inequality (mathematics), inequality and Monotonic function, monotonic functions. It is often att ...
, ''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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

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 Crystallography is the experimental science of determining the arrangement of atoms in crystalline so ...
of an -
simplex In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...
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 7-7-7 equilateral triangle would be approximately 21.22, which is larger than three times the area of a 7-4-4 isosceles triangle (approximately 6.78 each, by
Heron's formula In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

), and so the arrangement is forbidden by the tetrahedral inequality.

# Normed vector space

In a
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, one of the defining properties of the
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
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
subadditivityIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. For any proposed function to behave as a norm, it must satisfy this requirement. If the normed space is
euclidean Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of: Geometry *Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...
, 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 is a form of geometry in which the 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 of thei ...

) and denote and . Then the triangle formed by , , and , is non-degenerate but :$\, x+y\, =\, \left(1,1\right)\, =, 1, +, 1, =2=\, x\, +\, y\, .$

## Example norms

*''Absolute value as norm for the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
.'' To be a norm, the triangle inequality requires that the
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

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, :$-\left( \left\vert x \right\vert + \left\vert y \right\vert \right) \leq x+y \leq \left\vert x \right\vert + \left\vert y \right\vert$ 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 Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
.'' 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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. *: 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...
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 A parallelogram. The sides are shown in blue and the diagonals in red. 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 ...

. The triangle inequality for general values of is called Minkowski's inequality. It takes the form:$\, x+y\, _p \le \, x\, _p + \, y\, _p \ .$

# Metric space

In a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
with metric , the triangle inequality is a requirement upon
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

: :$d\left(x,\ z\right) \le d\left(x,\ y\right) + d\left(y,\ z\right) \ ,$ 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 An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
in a metric space is a
Cauchy sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 $d\left(x,\cdot\right)$ are
Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ex ...

with Lipschitz constant , and therefore are in particular
uniformly continuous In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. The proof for the reverse triangle uses the regular triangle inequality, and $\, y-x\, = \, 1\left(x-y\right)\, = , 1, \cdot\, x-y\, = \, x-y\,$: : $\, x\, = \, \left(x-y\right) + y\, \leq \, x-y\, + \, y\, \Rightarrow \, x\, - \, y\, \leq \, x-y\, ,$ : $\, y\, = \, \left(y-x\right) + x\, \leq \, y-x\, + \, x\, \Rightarrow \, x\, - \, y\, \geq -\, x-y\, ,$ 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclid ...
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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
. 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.

*
SubadditivityIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
*
Minkowski inequality In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, Fran ...
* Ptolemy's inequality

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