We study the evolution of a closed, convex hypersurface in Rⁿ⁺¹ in direction of its normal vector, where the speed equals a power k >= 1 of the mean curvature. We show that if initially the ratio of the biggest and smallest principal curvature at every point is close enough to 1, depending only on k and n, then this is maintained under the flow. As a consequence we obtain that, when rescaling appropriately as the flow contracts to a point, the evolving surfaces converge to the unit sphere.