In this paper we prove the following geometric inequality in the hyperbolic
space \H^n (nâ„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 ÎŁ is a horospherical convex
hypersurface. Equality holds if and only if ÎŁ is a geodesic sphere in
\H^n.Comment: 18page