# Flatness directions of the operator norm

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 $${\mathbb R}^n$$.

(You write $$\|\cdot\|_2$$ but say "operator norm", and I assume that is what you mean.)
Akemann and Pedersen show that the weakly closed faces of the unit ball of a von Neumann algebra are precisely the sets of the form $$v + (1 - vv^*)B(1 - v^*v)$$ where $$B$$ is the full unit ball and $$v$$ is a partial isometry. In finite dimensions all faces are weakly closed, of course. The difference of two elements of such a face would have the form $$(1 - vv^*)x(1 - v^*v) = pxq$$ where $$p$$ and $$q$$ are orthogonal projections. That can be nonsingular only if $$p = q = 1$$, which would imply $$v = 0$$, making the "face" be all of $$B$$ (if you consider that to be a face).
Your result is stated for real matrices, but any face of the unit ball of $$M_n(\mathbb{R})$$ is contained in a face of the unit ball of $$M_n(\mathbb{C})$$, so nothing more needs to be said.