Diameter and curvature control under mean curvature flow

We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^{n-1}$-estimates for the regularity scale of the level set flow with two-convex initial data. The results, which seem new even in the most classical case of mean convex surfaces evolving by mean curvature flow in $\Bbb{R}^3$, are ultimately a consequence of the Lojasiewicz inequality from Colding-Minicozzi.