# Triangle inequality

In

mathematics , the**triangle inequality**states that for anytriangle , 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**R**^{2}or**R**^{3}. The figure at the right shows two examples.The triangle inequality is a theorem in spaces such as the

real number s, allEuclidean space s, the L^{p}spaces ("p" ≥ 1), and anyinner product space . It also appears as an axiom in the definition of many structures inmathematical analysis andfunctional analysis , such asnormed vector space s andmetric space s.**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 theabsolute 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":

:$igg||x|-|y|igg|\; leq\; |x-y|.$

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") ≤ "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:: $igg||x|-|y|igg|\; leq\; |x-y|$, or for metric spaces, | "d"("x", "y") − "d"("x", "z") | ≤ "d"("y", "z"): $igg||x|-|y|igg|\; leq\; |x+y|$this implies that the norm ||–|| as well as the distance function "d"("x", –) 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 inspecial relativity .**ee also***

Subadditivity

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2010.*

### Look at other dictionaries:

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