Volume-preserving flow by powers of the m-th mean curvature ABSTRACT In this talk, we will analyse the evolution of closed hypersurfaces in the Euclidean space for which a contraction by a power of the m-th mean curvature (which can be regarded as a generalization of the mean, Gauss and scalar curvatures) is balanced by a spatially constant expansion to keep the volume fixed. We concentrate on velocities with degree of homogeneity greater than 1. Asking initially that the principal curvatures at each point lie in a suitably small cone about the umbilic line, we prove: (a) preservation of such a pinching condition, (b) existence of the solution for all time, and (c) exponential and smooth convergence to a round sphere.