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Minkowski Inequalities via Nonlinear Potential Theory

In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set $Ω\subset \mathbb R^n$, $n\geq 3$. Our proof relies on the discovery of effective monotonicity formulas holding along the level set flow of the $p$-capacitary potentials associated with $Ω$, for every $p$ sufficiently close to $1$. Besides constituting a neat improvement of those introduced in [Fog_Maz_Pin] to treat the case of convex domains, these formulas testify the existence of a link between the monotonicity formulas derived by Colding and Minicozzi for the level set flow of Green's functions and the monotonicity formulas employed by Huisken, Ilmanen and several other authors in studying the geometric implications of the Inverse Mean Curvature Flow. In dimension $n\geq 8$, our conclusions are stronger than the ones obtained so far through the latter mentioned technique.