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Piecewise smooth embedding of a cube

For unfoldings of non-negative curvature that are homeomorphic to the sphere S2, Aleksandrov [1] proved the existence of a realization of each of them as the surface of a convex polyhedron in R3 and the uniqueness of such a realization in the class of convex polyhedra. Olovyanishnikov [2] proved the uniqueness of the realization in the class of convex bodies.1 Besides the unique convex embedding in R3, there exist for each unfolding arbitrarily many non-convex polyhedral embeddings in each of which all faces are convex polygons [3]. In this paper we give an example of a piecewise smooth2 but not piecewise linear [4] embedding of the surface of a convex polyhedron in R3.