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 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.

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| ≤ |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 distance function d(x, -) are 1-Lipschitz and therefore continuous.

See also Cauchy-Schwarz inequality.

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 tx ty ≥ 0

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

de:Dreiecksungleichung es:Desigualdad del triángulo fr:Inégalité triangulaire it:Disuguaglianza triangolare he:אי שוויון המשולש nl:Driehoeksongelijkheid ja:三角不等式 pl:Nierówność trójkąta sv:Triangelolikheten zh:三角不等式