Some Bonnesen-style inequalities for higher dimensions. (English) Zbl 1272.52006
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.
Reviewer: Monika Nowicka (Bydgoszcz)
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
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