Reverse Triangle Inequality

Intoduction for Reverse Triangle Inquality:

Generally in mathematics, the triangle inequality states that the sum of length of any two sides is greater than the  length of remaining sides.

In the Euclidan geometry and also in some other geometries the triangle inequality is a theorem about distances. Euclidean geometry,has right triangle which is  a consequence of Pythagoras theorem. Thus, in a Euclidean geometry, the shortest distance between two points is a straight line.While in,the Spherical geometry  the shortest distance between two points is an arc of a circle.

Formula for Triangle Inequality:

The triangle inequality formula is shown in the following as,


Let x and y be the vectors that form two sides of the triangle in which the third side is x+y. The expression ||x|| denotes the length of a vector x. It is used ubiquitously throughout mathematics in reverse triangle inequality.
Example Problems of Reverse Triangular Inequality:


Let us consider if suppose without loss of generality is that ||x|| is no smaller than ||y||. (Else  we just interchange the roles of xy.)so,now the problem is: and


Thus we have to show that given statement is,

(x ) - (y)(x-y)

This is followed directly from the triangle inequality itself if we write the  x as the following,

X = x-y+y

And consider it as,

X = (x-y) + y.

By taking the norms and by applying the triangle inequality, it gives us,


Solve the following reverse triangle inequality with the formula mentioned above,

(a+b)`<=` (a)+(b)


We have to notice that,

-(a)`<=a``<=` (a)

and that can be written as,

-(b)`lt=` b`lt=` (b)

Now  by adding the two statements together, we get

a+b-(a+b)`<= ` a+b `<=` (a)+(b)

With help of  the property of absolute value, we will get that,

(a+b)`<=` (a)+(b)

Thus we are solved the above problem.