A new variational characterization of the Wulff shape

Given a positive function F   on SnSn which satisfies a convexity condition, we define the r  th anisotropic mean curvature function MrMr for hypersurfaces in Rn+1Rn+1 which is a generalization of the usual r  th mean curvature function. Let X:M→Rn+1 be an n  -dimensional closed hypersurface with Mr+1Mr=constant, for some r   with 1⩽r⩽n−11⩽r⩽n−1, which is a critical point for a variational problem. We show that X(M)X(M) is stable if and only if X(M)X(M) is the Wulff shape.