Vertex-rounding a three-dimensional polyhedral subdivision

Let P be a polyhedral subdivision in R sup 3 with a total of n faces. We show that there is an embedding sigma of the vertices, edges, and facets of P into a subdivision Q, where every vertex coordinate of Q is an integral multiple of 2 sup -[log sub 2 n+2]. For each face f of P, the Hausdorff distance in the L sub inf metric between f and sigma(f) is at most 3/2. The embedding sigma preserves or collapses vertical order on faces of P. The subdivision Q has O(n sup 4) vertices in the worst case , and can be computed in the same time.