From Minkowsky inequality for non-convex domains to an exotic functional inequality

Minkowsky inequality states that, among convex domains with the same perimeter of the ball, the integral of the absolute value of the mean curvature of the boundary is minimized by the ball. The validity of the inequality is open for general domains (and also for mean-convex domains).

Following Fuglede's proof of the stability of the isoperimetric inequality, one can show that Minkowsky inequality for C^1 perturbations of the ball is "almost equivalent" to an exotic functional inequality involving the Laplacian. Thanks to this "almost equivalence", one can prove some special cases and some variations of Minkowsky inequality for non-convex domains.