Integral formula of Minkowski type and new characterization of the Wulff shape

Abstract

Given a positive function F on Sn which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature Mr for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature Hr. We get integral formulas of Minkowski type for compact hypersurfaces in Rn+1. We give some new characterizations of the Wulff shape by use of our integral formulas of Minkowski type, in case F = 1 which reduces to some well-known results.