Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz continuous function with *strict* Lipschitz constant $L > 0$.

That is, $|f(x) - f(y)| < L|x - y|$ for all $x \neq y$ in $\mathbb R^d$.

**Question:** What is the maximal Hausdorff dimension of the set on which $f$ is differentiable and $|Df| = L$?

**Remarks:**

$\mathcal H^{n-1}$ is trivially achievable.

By sticking together a maximising sequence, we may achieve the supremal Hausdorff dimension, so we are justified in speaking of the maximum.