Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I

Abstract: In this paper we prove the following geometric inequality in the hyperbolic space $\H^n$ ($n\ge 5)$, which is a hyperbolic Alexandrov-Fenchel inequality, [\begin{array}{rcl} \ds \int_\Sigma \s_4 d \mu\ge \ds\vs C_{n-1}^4\omega_{n-1}\left{\left(\frac{|\Sigma|}{\omega_{n-1}} \right)^\frac 12 + \left(\frac{|\Sigma|}{\omega_{n-1}} \right)^{\frac 12\frac {n-5}{n-1}} \right}^2, \end{array}] provided that $\Sigma$ is a horospherical convex hypersurface. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\H^n$.