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In this paper we analyze the capacitary potential due to a charged body inorder to deduce sharp analytic and geometric inequalities, whose equality casesare saturated by domains with spherical symmetry. In particular, for a regularbounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 3$, we prove that if themean curvature $H$ of the boundary obeys the condition $$ - \bigg[\frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} \leq \frac{H}{n-1} \leq\bigg[ \frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} , $$ then $\Omega$is a round ball.

Some Sphere Theorems in Linear Potential Theory