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The general statement of the coarea inequality known also as the Eilenberg inequality is: Theorem. If $f:X\to Y$ is a Lipschitz map between metric spaces and $A\subset X$, $0\leq m\leq n$, then $$ \...

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 $\Omega$...