Triangle inequality: Definition from Answers.com

Two examples of the triangle inequality. The top example shows the case when there is a strict inequality and the second example shows the case when there is an equality.

In mathematics, the triangle inequality states that for any triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater than or equal to the difference between the two sides.

In Euclidean geometry and some other geometries this is a theorem. In the Euclidean case, in both the less than or equal to and greater than or equal to statements, equality occurs only if the triangle has a 180° angle and two 0° angles, as shown in the bottom example in the image to the right. The inequality can be viewed intuitively in either R² or R³. The figure at the right shows two examples.

The triangle inequality is a theorem in spaces such as the real numbers, all Euclidean spaces, the L^p spaces (p ≥ 1), and any inner product space. It also appears as an axiom in the definition of many structures in mathematical analysis and functional analysis, such as normed vector spaces and metric spaces.

1 Normed vector space
2 Metric space
3 Reverse triangle inequality
4 Reversal in Minkowski space
5 See also
6 References

Normed vector space

In a normed vector space V, the triangle inequality is

$\displaystyle \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V$

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.

The real line is a normed vector space with the absolute value as the norm, and so the triangle inequality states that for any real numbers x and y

$|x + y| \leq |x|+|y|.\,$

The triangle inequality is useful in mathematical analysis 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 x and y:

$|x-y| \geq \bigg||x|-|y|\bigg|.$

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 as follows: Given vectors x and y,

$\\|x + y\\|^2$	$= \langle x + y, x + y \rangle$
	$= \\|x\\|^2 + \langle x, y \rangle + \langle y, x \rangle + \\|y\\|^2$
	$\le \\|x\\|^2 + 2\|\langle x, y \rangle\| + \\|y\\|^2$
	$\le \\|x\\|^2 + 2\\|x\\|\\|y\\| + \\|y\\|^2$ (by the Cauchy-Schwarz Inequality)
	$= \left(\\|x\\| + \\|y\\|\right)^2$

Taking the square root of the final result gives the triangle inequality.

Metric space

In a metric space M with metric d, the triangle inequality is

d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z in M

that is, the distance from x to z is at most as large as the sum of the distance from x to y and the distance from y to z.

Reverse triangle inequality

The reverse triangle inequality is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds:

$\bigg|\|x\|-\|y\|\bigg| \leq \|x-y\|,$

or for metric spaces, | d(x, y) − d(x, z) | ≤ d(y, z). This implies that the norm ||–|| as well as the distance function d(x, –) are Lipschitz continuous with Lipschitz constant 1, and therefore are in particular continuous.

Reversal in Minkowski space

In the usual Minkowski space and in Minkowski space extended to an arbitrary number of spatial dimensions, assuming null or timelike vectors in the same time direction, the triangle inequality is reversed:

$\|x+y\| \geq \|x\| + \|y\| \; \forall x, y \in V$ such that $\|x\|, \|y\| \geq 0$ and $t_x , t_y \geq 0$ .

A physical example of this inequality is the twin paradox in special relativity.

References

Pedoe, Daniel (1988), Geometry: A comprehensive course, Dover, ISBN 0-486-65812-0 .
Rudin, Walter (1976), Principles of Mathematical Analysis, New York: McGraw-Hill, ISBN 007054235X .

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$\\|x + y\\|^2$	$= \langle x + y, x + y \rangle$
	$= \\|x\\|^2 + \langle x, y \rangle + \langle y, x \rangle + \\|y\\|^2$
	$\le \\|x\\|^2 + 2\|\langle x, y \rangle\| + \\|y\\|^2$
	$\le \\|x\\|^2 + 2\\|x\\|\\|y\\| + \\|y\\|^2$ (by the Cauchy-Schwarz Inequality)
	$= \left(\\|x\\| + \\|y\\|\right)^2$

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Triangle inequality

Contents

Normed vector space

Metric space

Reverse triangle inequality

Reversal in Minkowski space

See also

References