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Recently, the 1st named author together, with Xinan Ma [ 12 ], has 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 for uniformly convex domains in   $\mathbb {R}^{n}$  . 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 by Reilly [ 18 ].

Classical Neumann Problems for Hessian Equations and Alexandrov–Fenchel’s Inequalities