Compact Embedded Hypersurfaces with Constant Higher Order Anisotropic Mean Curvatures

Given a positive function F on S-n which satisfies a convexity condition, for 1 <= r <= n, we define for hypersurfaces in Rn+1 the r-th anisotropic mean curvature function H-r(F), a generalization of the usual r-th mean curvature function. We prove that a compact embedded hypersurface without boundary in Rn+1 with H-r(F) constant is the Wulff shape, up to translations and homotheties. In the case r = 1, our result is the anisotropic version of Alexandrov's Theorem, which gives an affirmative answer to an open problem of E Morgan.