On MathOverflow I saw this inequality. Let $E$ is a normed linear space.
$$ \|x+y\|+\|y+z\|+\|z+x\|\le\|x\|+\|y\|+\|z\|+\|x+y+z\|,\qquad\forall x,y,z\in E $$
Apparently this is always true if $E = \mathbb{R}^n$. I always thought Triangle inequality was the only foundation inequality for normed spaces. Does Hlawka inequality follow as a consequence?
The same MO question discusses potential counterexamples. Why is the space of $2 \times 2$ self-adjoint matrices not Euclidean with the following norm?
$$ ||A|| = \tfrac{1}{2}|\mathrm{tr}( A)| + \tfrac{1}{\sqrt{2}}\big|\big| A - \mathrm{tr}(A)I_{2\times 2} \big|\big|_2$$
The weighted sum of the trace and the Hilbert-Schmidt norm.
