Triangle inequality

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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.
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 is the theorem stating 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 both the less than or equal to and greater than or equal to statements, equality only occurs in the case of a triangle that 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 ≥ 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.

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[edit] 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:

\bigg||x|-|y|\bigg| \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.

[edit] 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.

[edit] Proof

Using the Cauchy-Schwarz Inequality, the Triangle Inequality is proved generally for any well defined inner product space 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
= \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2
\le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2
\le \left(\|x\| + \|y\|\right)^2

where the Cauchy-Schwarz Inequality is used in the fourth line. Taking the square root of the final result gives the triangle inequality.

[edit] Consequences

The following consequences of the triangle inequalities are often useful; they give 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)
\bigg|\|x\|-\|y\|\bigg| \leq \|x+y\|

this implies that the norm ||–|| as well as the distance function d(x, –) are 1-Lipschitz and therefore continuous.

[edit] 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.

[edit] See also