We consider convex hypersurfaces for which the ratio of principal curvaturesat each point is bounded by a function of the maximum principal curvature withlimit 1 at infinity. We prove that the ratio of circumradius to inradius isbounded by a function of the circumradius with limit 1 at zero. We apply thisresult to the motion of hypersurfaces by arbitrary speeds which are smoothhomogeneous functions of the principal curvatures of degree greater than one.For smooth, strictly convex initial hypersurfaces with ratio of principalcurvatures sufficiently close to one at each point, we prove that solutionsremain smooth and strictly convex and become spherical in shape whilecontracting to points in finite time.
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