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

By applying the unit normal flow to well-known inequalities in hyperbolic space n+1 and in the sphere Sn+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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isoperimetric inequality, convex hypersurface, Alexandrov–Fenchel-type inequality, $k$-th order mean curvature, Gauss–Bonnet curvature
Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 53C42
Received: 6 October 2013
Revised: 8 May 2014
Accepted: 10 March 2015
Published: 6 August 2015
Yong Wei
Department of Mathematical Sciences
Tsinghua University
100084, Beijing
Department of Mathematics
University College London
London WC1E 6BT
United Kingdom
Changwei Xiong
Department of Mathematical Sciences
Tsinghua University
100084, Beijing