Triangle inequality

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In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides.

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.

Contents

1 Normed vector space
2 Metric space
3 Consequences
4 Reversal in Minkowski space

Normed vector space

In a normed vector space V, the triangle inequality is

||x + y|| ≤ ||x|| + ||y|| for all 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.

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| \le |x|+|y|.]

The triangle inequaltiy 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:

[\Big| |x|-|y| \Big| \le |x+y|.]

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.

Consequences

The following consequences of the triangle inequalities are often useful; they give lower bounds instead of upper bounds:

| ||x|| - ||y|| | ≤ ||x - y|| or for metric | d(x, y) - d(x, z) | ≤ d(y, z)

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|| ≥ ||x|| + ||y|| for all x, y in V such that ||x|| ≥ 0, ||y|| ≥ 0 and t_x t_y ≥ 0

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

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