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This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $\beta (\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one 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. This result covers and generalises the previous results for convex hypersurfaces in the Euclidean space by McCoy \cite{McC05} and Cabezas-Rivas and Sinestrari \cite{CS10} to more general curvature flows for convex hypersurfaces with similar curvature pinching condition.

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