Asymptotically Efficient Triangulations of the D-Cube

Abstract

Let P and Q be polytopes, the first of "low" dimension and the second of "high" dimension. We show how to triangulate the product P × Q efficiently (i.e., with few simplices) starting with a given triangulation of Q. Our method has a computational part, where we need to compute an efficient triangulation of P × Δm, for a (small) natural number m of our choice. Δm denotes the m-simplex. Our procedure can be applied to obtain (asymptotically) efficient triangulations of the cube In : We decompose In = I k × In-k, for a small k. Then we recursively assume we have obtained an efficient triangulation of the second factor and use our method to triangulate the product. The outcome is that using k = 3 and m = 2, we can triangulate In with O(0.816nn!) simplices, instead of the O(0.840nn!) achievable before.