Reverse Triangle Inequality

The reverse triangle inequality, sometimes called the circle inequality (see below) is a simple consequence of the triangle inequality.

Proof

We first consider the norm of $\mathit{x}$ expressed in terms of $\mathit{y}$, and apply the triangle inequality:

This implies that:

Since $\mathit{x}$ and $\mathit{y}$ are generic, we also have that:

Combining these results we have the desired inequality:

Geometric Intuition

This inequality is also sometimes called the circle inequality due to the below interpretation in ${\mathbb{R}}^2$ with the Euclidean norm or $\mathbb{C}$ with the complex modulus.

Given two complex numbers $z$ and $w$, with two circles drawn for the set of points with modulus equal to that of $z$ and $w$ each. Then, $|z-w|$ is the distance between a point on the inner circle and a point on the outer circle, while $||z|-|w||$ is the difference between the radii of the two circles. The latter is the minimum of the former.