Prove that $\begin{align*}&|a+b||a+c|+|a+b||b+c|+|a+c||b+c|\\ \leq &(|a|+|b|+|c|) \cdot |a+b+c|+|a||b|+|a||c|+|b||c|\end{align*}$ in Euclidean space $\mathbb{R}^n$.
I have been thinking about this inequality for 2 weeks. This exercise was in my exam in functional analysis. I think, we have to use fact, that $|x+y|^2=\langle x,x\rangle+2\langle x,y\rangle +\langle y,y\rangle$ and $|x+y|\leq |x|+|y|$ and symmetries properties, but I can not find a good proof.
If $a,b,c\in \mathbb{R}$ then it is easy to prove this inequality. We have to prove following inequalities $|a+b||a+c|\leq |a||a+b+c|+|b||c|$ $|a+b||b+c|\leq |b||a+b+c|+|a||c|$ $|a+c||b+c|\leq |c||a+b+c|+|a||b|$ There is symetry therofore we can prove only one equation. It is easy to show that $|a+b||a+c|=|a(a+b+c)+bc|\leq |a||a+b+c|+|b||c|$ We take the sum of 3 inequalities above and the proof is ended.
I was trying to prove inequality analogues in the space $\mathbb{R^n}$, but without a success. If I take a square of one of the inequalities, I can not get simplifier inequality.
P.S. Please, correct my grammar mistakes