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

DOI: 10.2140/pjm.2015.277.219

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

Mathematical Subject Classification 2010

Primary: 53C44

Secondary: 53C42

Milestones

Received: 6 October 2013

Revised: 8 May 2014

Accepted: 10 March 2015

Published: 6 August 2015