Sharp geometric quantitative estimates

Abstract

Let $E\subset B\subset {\mathbb{R}}^n$ be closed, bounded, convex sets. The monotonicity of the surface areas tells us that

We give quantitative estimates from below of the difference $\delta (E;B)={\mathcal{H}}^{n-1}(\partial B)-{\mathcal{H}}^{n-1}(\partial E)$ in the cases $n=2$ and $n=3$. As an application, considered a decomposition of a closed and bounded set into a number $k$ of convex pieces, we deduce an estimate from below of the minimal number of convex components that may exist.