#### Vol. 277, No. 1, 2015

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Inequalities of Alexandrov–Fenchel type for convex hypersurfaces in hyperbolic space and in the sphere

### Yong Wei and Changwei Xiong

Vol. 277 (2015), No. 1, 219–239
##### Abstract

By applying the unit normal flow to well-known inequalities in hyperbolic space ${ℍ}^{n+\mathfrak{1}}$ and in the sphere ${\mathbb{S}}^{n+\mathfrak{1}}$, we derive some new inequalities of Alexandrov–Fenchel type for closed convex hypersurfaces in these spaces. We also use the inverse mean curvature flow in the sphere to prove an optimal Sobolev-type inequality for closed convex hypersurfaces in the sphere.

##### Keywords
isoperimetric inequality, convex hypersurface, Alexandrov–Fenchel-type inequality, $k$-th order mean curvature, Gauss–Bonnet curvature
Primary: 53C44
Secondary: 53C42