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We consider a convex Euclidean hypersurface that evolves by a volume- or area-preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the monotonicity of the isoperimetric ratio, which allows to control the inner radius and outer radius of the hypersurface and to deduce uniform bounds on the curvature by maximum principle arguments.

Volume-preserving nonhomogeneous mean curvature flow of convex hypersurfaces