Hlawka’s inequality

Theorem 1

In an inner product space (http://planetmath.org/InnerProductSpace), let $x\mathrm{,}y\mathrm{,}z$ be vectors. Then

$$\parallel x+y\parallel +\parallel y+z\parallel +\parallel z+x\parallel \le \parallel x\parallel +\parallel y\parallel +\parallel z\parallel +\parallel x+y+z\parallel .$$