A sharp quantitative estimate for the surface areas of convex sets in ${\mathbb{R}}^3$

Abstract

Let $E\subset B\subset {\mathbb{R}}^3$ be closed, bounded, convex sets. It is known that the monotonicity of the surface areas holds, i.e. ${\mathcal{H}}^2(\partial E)⩽{\mathcal{H}}^2(\partial B)$. Here we give a quantitative estimate of the difference of the surface areas from below depending on the Hausdorff distance between $E$ and $B$. Moreover, we construct an example which shows the sharpness of our result.