# Triangle inequality

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 Lp spaces ("p" &ge; 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.

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 "inverse triangle inequality" which states that for any real numbers "x" and "y":

:

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.

Metric space

In a metric space "M" with metric "d", the triangle inequality is: "d"("x", "z") &le; "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".

Proof

The triangle inequality is proved generally for any well defined inner product space as follows:

Given vectors "x" and "y",:Taking the square root of the final result gives the triangle inequality.

Consequences

The following consequences of the triangle inequalities are often useful; they give lower bounds instead of upper bounds:: , or for metric spaces, | "d"("x", "y") − "d"("x", "z") | &le; "d"("y", "z"): this implies that the norm ||&ndash;|| as well as the distance function "d"("x", &ndash;) are 1-Lipschitz and therefore 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.

ee also

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• triangle inequality — noun The inequality that states that the magnitude of the sum of two vectors is less than or equal to the sum of the magnitudes of the vectors, or any equivalent inequality in other spaces …   Wiktionary

• triangle inequality — noun Etymology: from its application to the distances between three points in a coordinate system Date: 1941 an inequality stating that the absolute value of a sum is less than or equal to the sum of the absolute values of the terms …   New Collegiate Dictionary

• triangle inequality — noun Etymology: so called from its application to the distances between three points in a coordinate system : an inequality stating that the absolute value of a sum is less than or equal to the sum of the absolute value of the terms * * * Math. 1 …   Useful english dictionary

• triangle inequality — Math. 1. the theorem that the absolute value of the sum of two quantities is less than or equal to the sum of the absolute values of the quantities. 2. the related theorem that the magnitude of the sum of two vectors is less than or equal to the… …   Universalium

• Triangle — This article is about the basic geometric shape. For other uses, see Triangle (disambiguation). Isosceles and Acute Triangle redirect here. For the trapezoid, see Isosceles trapezoid. For The Welcome to Paradox episode, see List of Welcome to… …   Wikipedia

• Inequality — In mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not (See also: equality) *The notation a < b means that a is less than b . *The notation a > b means that a is… …   Wikipedia

• inequality — /in i kwol i tee/, n., pl. inequalities. 1. the condition of being unequal; lack of equality; disparity: inequality of size. 2. social disparity: inequality between the rich and the poor. 3. disparity or relative inadequacy in natural endowments …   Universalium

• Inequality (mathematics) — Not to be confused with Inequation. Less than and Greater than redirect here. For the use of the < and > signs as punctuation, see Bracket. More than redirects here. For the UK insurance brand, see RSA Insurance Group. The feasible regions… …   Wikipedia

• Triangle (disambiguation) — A triangle is a geometric shape that has three straight sides. Triangle can also refer to: Mathematics * Spherical triangle * Sierpinski triangle * Pascal s triangle * Triangle wave * Triangle inequality Objects * Triangle (instrument), a musical …   Wikipedia

• List of triangle topics — This list of triangle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as Pascal s triangle or triangular matrices, or concretely in physical space.… …   Wikipedia