Subdivision algorithms and the kernel of a polyhedron (Q1196202)

The Fejes-Tóth kernel of a complex polytope is the limiting set obtained when iteratively taking the convex hull of the midpoints of all edges (midpoint process). It is shown that in dimension 3 this kernel has no 2-dimensional faces, that the centroids of all facets obtained during the iterations is dense in the kernel's boundary, and that this boundary may not be twice continuously differentiable. The midpoint process is related to smoothing techniques used in CAD, thereby indicating limits to the smoothing obtained by such techniques in dimension 3. For another smoothing process defined by DOO, it is shown that the corresponding kernel may be nonconvex.