In this paper we consider convex hypersurfaces in anisotropic geometries determined by a smooth, strictly convex Wulff shape. We consider the volume preserving flow of smooth closed convex hypersurfaces in the Euclidean space with speed in the anisotropic normal direction given by a positive power \(\alpha \) of the kth anisotropic mean curvature plus a global term chosen to preserve the enclosed volume of the evolving hypersurfaces. We apply a result of Daniel Hug which characterises the Wulff shape using anisotropic curvature measures to prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a scaled translate of the Wulff shape in the Hausdorff sense. Moreover, under some additional assumptions, we can further improve the Hausdorff convergence to smooth convergence at an exponential rate.
