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Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$. Let $\Omega$...

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $\nabla f$ nonzero almost everywhere with respect to Lebesgue measure. What is the supremal Hausdorff dimension of the set on which $f$ ...