Mean curvature flow of star-shaped hypersurfaces

In the last 15 years, the series of works of White and Huisken-Sinestrari yield that the blowup limits at singularities are convex for the mean curvature flow of mean convex hypersurfaces. In 1998 Smoczyk [20] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in R-3. We prove in this paper that this is still true for the mean curvature flow of star-shaped hypersurfaces in Rn+1 in arbitrary dimension n >= 2. In fact, this holds for a much more general class of initial hypersurfaces. In particular, this implies that the mean curvature flow of star-shaped hypersurfaces is generic in the sense of Colding-Minicozzi [6].