I am trying to prove the next inequality:
Let $a, b, c, d ∈ \mathbb{R}^n.$ Then $$||a − c|| · ||b − d|| ≤ ||a − b|| · ||c − d|| + ||a − d|| · ||b − c||.$$
I was considering function $f(x,y,z)= ||x|| · ||y − z|| + ||y|| · ||x − y||- ||y|| · ||x − z||$ which is defined in $\mathbb{R}^{3n},$ and then to extended with the case when $a\neq 0.$
Such function is differentianble, so I was computing its gradient to get a critic point, but each coordinate is a complicated expresion.
As given hint is the next: Reduce the problem to the case a = 0 and consider the mapping $ϕ(x) =\frac {x}{||x||^2}$ for $x\neq 0.$ When does equality hold?
I cannot see why the hint above is useful if we have a function from $\mathbb{R}^n$ to $\mathbb{R}^n$ if we want to achieve a minimum, I guess, of a function who gives the desire inequality.
Any kind of help is thanked in advanced.