Sharp geometric quantitative estimates (Q524831)

Summary: Let \(E \subset B \subset \mathbb{R}^n\) be closed, bounded, convex sets. The monotonicity of the surface areas tells us that \[ \mathcal{H}^{n-1}(\partial E) \leq \mathcal{H}^{n-1}(\partial B) \] where \(\mathcal{H}^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure. 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.