It is known that the standard operator norm $\|\cdot\|_2$ over ${\bf M}_n({\mathbb R})$ is very flat, as is any operator norm (= subordinated norm) actually. The set of extremal points of the unit ball is the very small set ${\bf O}_n({\mathbb R})$ and the unit sphere contains faces (convex subsets) of dimension $(n-1)^2$.

I discovered that the directions of flatness actually are singular matrices: if a segment $[M,N]$ is included in the unit sphere, then $\det(M-N)=0$. I hardly pretend that this is an original result.

Is there any reference in the literature for the above result ?

I should be also interested in any related statement for other operator norms.

**Notation**. I use to write $\|\cdot\|_p$ for the norm over ${\bf M}_n({\mathbb R})$ induced by the $\ell^p$ norm over ${\mathbb R}^n$.