Convex hypersurfaces evolving by volume preserving curvature flows

We consider the evolution of a closed convex hypersurface in euclidean space under a volume preserving flow whose speed is given by a positive power of the mean curvature. We prove that the solution exists for all times and converges to a sphere. The result does not assume the curvature pinching properties or the restrictions on the dimension that were usually required in the previous literature. The proof of the convergence exploits the monotonicity of the isoperimetric ratio satisfied by this class of flows.