Let $E$ be a finite dimensional normed vector space. If $E$ is $\ell^1$-embeddable, then the norm satisfies Hlawka inequality
$${\bf(H)}\qquad\|x+y\|+\|y+z\|+\|z+x\|\le\|x\|+\|y\|+\|z\|+\|x+y+z\|,\qquad\forall x,y,z\in E.$$
This is how one can prove that every Euclidian space satisfies **(H)**.


Now consider the Euclidian space ${\mathbb R}^n$.



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> Does $E={\mathcal L}({\mathbb R}^n)$, endowed with the operator norm, satisfy **(H)** ? If so, is it $\ell^1$-embeddable ?
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**Edit.** After Mateusz' answer, the two-dimensional case remains open.



Operator norm versus Hlawka inequality
It does not hold, at least for $n \geq 3$. The reason is that $M\_n$ contains then a subspace isometric to $\ell\_{\infty}^3$ (diagonal matrices with three non-zero entries) and for this space the inequality does not hold -- take $x=(1,1,0)$, $y=(0,1,1)$ and $z=(1,0,1)$.


