Volume preserving flow by powers of κ-TH mean curvature

We consider the flow of closed convex hypersurfaces in Euclidean space Rⁿ⁺¹with speed given by a power of the k-th mean curvature E_kplus a global term chosen to impose a constraint involving the enclosed volume V_n+1and the mixed volume V_n+1-kof 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.