  We study a volume preserving curvature flow of convex hypersurfaces, driven
by a power of the $k$-th elementary symmetric polynomial in the principal
curvatures. Unlike most of the previous works on related problems, we do not
require assumptions on the curvature pinching of the initial datum. We prove
that the solution exists for all times and that the speed remains bounded and
converges to a constant in an integral norm. In the case of the volume
preserving scalar curvature flow, we can prove that the hypersurfaces converge
smoothly and exponentially fast to a round sphere.
