  In this article, we extend Huisken's theorem that convex surfaces flow to
round points by mean curvature flow. We construct certain classes of mean
convex and non-mean convex hypersurfaces that shrink to round points and use
these constructions to create pathological examples of flows. We find a
sequence of flows that exist on a uniform time interval, have uniformly bounded
diameter, and shrink to round points, yet the sequence of initial surfaces has
no subsequence converging in the Gromov-Hausdorff sense. Moreover, we find a
sequence of flows which all shrink to round points, yet the initial surfaces
converge to a space-filling surface. Also constructed are surfaces of
arbitrarily large area which are close in Hausdorff distance to the round
sphere yet shrink to round points.
