Is there a nowhere differentiable norm on $\Bbb R^n$?

Question

Is there a nowhere differentiable norm on $\Bbb R^n$?

The non differentiable norms that I know (e.g. the one norm $\|\cdot\|_1$ and the infinity norm $\|\cdot\|_{\infty}$) are non differentiable at a few points (basically the corners of the associated unit ball) but I can't find/build any example of norm which is differentiable nowhere...

Answer 1

Rademacher's theorem: a locally Lipschitz function on a nonempty open subset of $\mathbb R^n$ is (Frechet) differentiable almost everywhere.