  In this paper, firstly, inspired by Nat\'{a}rio's recent work \cite{Na}, we
use the isoperimetric inequality to derive some Alexandrov-Fenchel type
inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$
and in the sphere $\SS^{n+1}$. We also get the rigidity in the spherical case.
Secondly, we use the inverse mean curvature flow in sphere \cite{gerh,Mak-Sch}
to prove an optimal Sobolev type inequality for closed convex hypersurfaces in
the sphere.
