  We consider the evolution of a closed convex hypersurface under a volume
preserving curvature flow. The speed is given by a power of the m-th mean
curvature plus a volume preserving term, including the case of powers of the
mean curvature or of the Gauss curvature. We prove that if the initial
hypersurface satisfies a suitable pinching condition, the solution exists for
all times and converges to a round sphere.
