triangle inequality

Let (X,d) be a metric space. The triangle inequalityMathworldMathworldPlanetmathPlanetmath states that for any three points x,y,zX we have


The name comes from the special case of n with the standard topology, and geometrically meaning that in any triangle, the sum of the lengths of two sides is greater (or equal) than the third.

Actually, the triangle inequality is one of the properties that define a metric, so it holds in any metric space. Two important cases are with d(x,y)=|x-y| and with d(x,y)=x-y (here we are using complex modulusMathworldPlanetmath, not absolute valueMathworldPlanetmath).

There is a second triangle inequality, sometimes called the reverse triangle inequality, which also holds in any metric space and is derived from the definition of metric:


In Euclidean geometryMathworldPlanetmath, this inequalityMathworldPlanetmath is expressed by saying that each side of a triangle is greater than the difference of the other two.

The reverse triangle inequality can be proved from the first triangle inequality, as we now show.

Let x,y,zX be given. For any a,b,cX, from the first triangle inequality we have:


and thus (using d(b,c)=d(c,b) for any b,cX):

d(a,c)d(a,b)-d(b,c) (1)

and writing (1) with a=x,b=z,c=y:

d(x,y)d(x,z)-d(z,y) (2)

while writing (1) with a=y,b=z,c=x we get:



d(x,y)d(z,y)-d(x,z); (3)

from (2) and (3), using the properties of the absolute value, it follows finally:


which is the second triangle inequality.

Title triangle inequality
Canonical name TriangleInequality
Date of creation 2013-03-22 12:14:49
Last modified on 2013-03-22 12:14:49
Owner drini (3)
Last modified by drini (3)
Numerical id 12
Author drini (3)
Entry type Definition
Classification msc 54-00
Classification msc 54E35
Related topic ProofOfLimitRuleOfProduct
Related topic TriangleInequalityOfComplexNumbers
Defines reverse triangle inequality