#### 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.

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##### Keywords
isoperimetric inequality, convex hypersurface, Alexandrov–Fenchel-type inequality, $k$-th order mean curvature, Gauss–Bonnet curvature
Primary: 53C44
Secondary: 53C42
##### Milestones
Received: 6 October 2013
Revised: 8 May 2014
Accepted: 10 March 2015
Published: 6 August 2015
##### Authors
 Yong Wei Department of Mathematical Sciences Tsinghua University 100084, Beijing China Department of Mathematics University College London London WC1E 6BT United Kingdom Changwei Xiong Department of Mathematical Sciences Tsinghua University 100084, Beijing China