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

This paper concerns the evolution of a closed hypersurface dimension [Formula: see text] in Euclidean space under mixed volume preserving flow. The speed equals power homogeneous curvature functions degree one and either convex or concave plus term, including case powers mean Gauss curvature. main result is that if initial satisfies suitable pinching condition, there exists unique, smooth solution flow for all times, evolving hypersurfaces converge exponentially to round sphere, enclosing same as hypersurface.