OPTIMAL SMOOTHING FOR CONVEX POLYTOPES - GHOMI, MOHAMMAD (The Bulletin of the London Mathematical Society, 2004)
It is proved that, given a convex polytope P in Rn, together with a collection of compact convex subsets in the interior of each facet of P, there exists a smooth convex body arbitrarily close to P that coincides with each facet precisely along the prescribed sets, and has positive curvature elsewhere. 2000 Mathematics Subject Classification 53A07, 52B11, 53C45.