  We present a new proof of the Willmore inequality for an arbitrary bounded
domain $\Omega\subset\mathbb{R}^{n}$ with smooth boundary. Our proof is based
on a parametric geometric inequality involving the electrostatic potential for
the domain $\Omega$; this geometric inequality is derived from a geometric
differential inequality in divergence form. Our parametric geometric inequality
also allows us to give new proofs of the quantitative Willmore-type and the
weighted Minkowski inequalities by Agostiniani and Mazzieri.
