If $E$ is a convex subset of $\mathbb{R}^n$ with $|E| = |B_1|$, then one consequence of the classical Alexandrov-Fenchel inequalities from convex geometry is that $$\int_{\partial E} H_{\partial E} \,ds \geq \int_{\partial B_1} H_{\partial B_1}\,ds,$$ where $H_{\Sigma}$ denotes the mean curvature of a surface $\Sigma$ (normalized, say, so that $H_{\partial B_1} = n-1$).

Can one generalize this inequality to sets that are not convex, but "semi convex" in the sense that all the boundary points have an exterior tangent ball of radius $r$? One candidate is: $$\int_{\partial E}\left( H_{\partial E} + \frac{n-1}{r}\right) \,ds \geq (?) \int_{\partial B_1}\left( H_{\partial B_1} + \frac{n-1}{r}\right) \,ds.$$ In the plane there is an easy proof using Gauss-Bonnet and the isoperimetric inequality. There doesn't seem to be an obvious counterexample in higher dimensions.