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.
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Mohammad Ghomi (2004) studied this question.