Quantitative oscillation estimates for almost-umbilical closed   hypersurfaces in Euclidean space
  We prove $\epsilon$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is $\delta$-small compared to the mean curvature. We give the explicit dependence of $\delta$ on $\epsilon$ within the class of uniformly convex hypersurfaces with bounded volume. 
Julian Scheuer
math.DG