Triangle inequality
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In mathematics, the triangle inequality states that the distance from A to B to C is never shorter than going directly from A to C. The triangle inequality is a theorem in spaces such as the real numbers, Euclidean space, Lp spaces (p ≥ 1) and in all inner product spaces; it is an axiom in the definition of abstract concepts such as normed vector spaces and metric spaces.
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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.
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 he:אי שוויון המשולש ja:三角不等式 pl:Nierówność trójkąta zh:三角不等式