Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Question

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Gentle hints only, please!

I know that attempting to decompose R.H.S. into

$$\alpha a + \beta b + \gamma c + \delta (a+b+c) = a + b$$

so that

$$\alpha \|a\| + \beta \|b\| + \gamma \|c\| + \delta \|a+b+c\| \geq \|a + b\|$$

and sum over all L.H.S. terms does not work.

I also know that I can interpret $\|a+b\|+\|b+c\|+\|c+a\|$ as the "straighter" path from $0$ to $2a+2b+2c$.

However, I haven't been able to translate that intuition into a proof! Hints only please!

Answer 1

Note that $(|a|+|b|+|c|-|b+c|-|a+c|-|a+b|+|a+b+c|)(|a|+|b|+|c|+|a+b+c|)=(|b|+|c|-|b+c|)(|a|-|b+c|+|a+b+c|)+(|c|+|a|-|c+a|)(|b|-|c+a|+|a+b+c|)+(|a|+|b|-|a+b|)(|c|-|a+b|+|a+b+c|)$

By done!