  We consider the flow of closed convex hypersurfaces in Euclidean space
$\mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature
$E_k$ plus a global term chosen to impose a constraint involving the enclosed
volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface.
We prove that if the initial hypersurface is strictly convex, then the solution
of the flow exists for all time and converges to a round sphere smoothly. No
curvature pinching assumption is required on the initial hypersurface.
