  Recently, the first named author together with Xinan Ma \cite{ma2015neumann},
have proved the existence of the Neumann problems for Hessian equations. In
this paper, we proceed further to study classical Neumann problems for Hessian
equations. We prove here the existence of classical Neumann problems under the
uniformly convex domain in R^n. As an application, we use the solution of the
classical Neumann problem to give a new proof of a family of Alexandrov-Fenchel
inequalities arising from convex geometry. This geometric application is
motivated from Reilly \cite{Reilly1980}.
