Mixed volume preserving flow by powers of homogeneous curvature functions of degree one

This paper concerns the evolution of a closed hypersurface of dimension n(>= 2) in the Euclidean space Double-struck capital Rn+1 under a mixed volume preserving flow. The speed equals a power beta(>= 1) of homogeneous curvature functions of degree one and either convex or concave plus a mixed volume preserving term, including the case of powers of the mean curvature and of the Gauss curvature. The main result is that if the initial hypersurface satisfies a suitable pinching condition, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces converge exponentially to a round sphere, enclosing the same mixed volume as the initial hypersurface.