On the exact maximum complexity of Minkowski sums of convex polyhedra

We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R3. In particular, we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with m and n facets respectively is bounded from above by f(m, n) = 4mn-9m-9n+26. Given two positive integers m and n, we describe how to construct two convex polyhedra with m and n facets respectively, such that the number of facets of their Minkowski sum is exactly f(m, n). We generalize the construction to yield a lower bound on the maximum complexity of Minkowski sums of many convex polyhedra in R3. That is, given k positive integers m1,m2, . . .,mk, we describe how to construct k convex polyhedra with corresponding number of facets, such that the number of facets of their Minkowski sum is ∑ 1≤i<j≤k (2m1-5) (2mj-5) +k2 +∑ 1 ≤ <k m1. We also provide a conservative upper bound for the general case. Snapshots of several pre-constructed convex polyhedra, the Minkowski sum of which is maximal, are available at http://www.cs.tau.ac.il/~efif/Mink. The polyhedra models and an interactive program that computes their Minkowski sums and visualizes them can be downloaded as well.