An inequality for inner product space: $\|a+b\|+\|b+c\|+\|c+a\|\le\|a\|+\|b\|+\|c\|+\|a+b+c\|$

Question

In an inner product space show that the following inequality holds: $$\|a+b\|+\|b+c\|+\|c+a\|\le\|a\|+\|b\|+\|c\|+\|a+b+c\|$$ I figured LHS and RHS are two different upper bounds on $\|2a+2b+2c\|$ using the triangle inequality in two different ways, but I can neither prove it, nor find anything like it.

Answer 1

This is Hlawka's inequality. We have to show that $$\|a\|+\|b\|+\|c\|-\|a+b\|-\|b+c\|-\|c+a\|+\|a+b+c\|\geq 0.$$ We multiply both sides by $\|a\|+\|b\|+\|c\|+\|a+b+c\|>0$ (if it is zero the inequality is trivial). We note that in a inner product space we have $$\|a+b+c\|^2+\|a\|^2+\|b\|^2+\|c\|^2=\|a+b\|^2+\|b+c\|^2+\|c+a\|^2.$$ Thus, we add $$\|a+b\|^2+\|b+c\|^2+\|c+a\|^2-\|a+b+c\|^2-\|a\|^2-\|b\|^2-\|c\|^2\geq 0$$ to our product and after some algebraic manipulation it reduces to \begin{align*} &(\|a\|+\|b\|-\|a+b\|)(\|a+b+c\|+\|c\|-\|a+b\|)\\ &+(\|b\|+\|c\|-\|b+c\|)(\|a+b+c\|+\|a\|-\|b+c\|)\\ &+(\|c\|+\|a\|-\|c+a\|)(\|a+b+c\|+\|b\|-\|c+a\|)\geq 0 \end{align*} which holds because, by the triangle inequality, in each of the three terms, both factors are not negative.

Answer 2

Hint: Since both sides are positive reals, it suffices to show $(\text{LHS})^2\leq(\text{RHS})^2.$