Some Bonnesen-style inequalities for higher dimensions
The authors consider the higher dimensional Bonnesen-style inequalities. They prove the following: If \(K\) is a domain in the Euclidean space \(\mathbb{R}^n\) with surface area \(S\) and volume \(V\) and \(K^{*}\) is the convex hull of \(K\) of the surface area \(S^{*}\) and volume \(V^{*}\), then for \(S\geq S^{*}\) \[ S^n-n^n \omega_nV^{n-1} \geq (S-S^{*})^n \quad\text{ and }\quad S^n-n^n \omega_nV^{n-1} \geq C \omega_n (V^{*}-V)^{n-1}, \] where \(C\) is a constant and \(\omega_n\) is the volume of the unit ball of \(\mathbb{R}^n\). Each equality holds when \(K\) is an \(n\)-ball.
scientific article; zbMATH DE number 6143614
scientific article