  We consider convex hypersurfaces for which the ratio of principal curvatures
at each point is bounded by a function of the maximum principal curvature with
limit 1 at infinity. We prove that the ratio of circumradius to inradius is
bounded by a function of the circumradius with limit 1 at zero. We apply this
result to the motion of hypersurfaces by arbitrary speeds which are smooth
homogeneous functions of the principal curvatures of degree greater than one.
For smooth, strictly convex initial hypersurfaces with ratio of principal
curvatures sufficiently close to one at each point, we prove that solutions
remain smooth and strictly convex and become spherical in shape while
contracting to points in finite time.
