Maximal f-vectors of Minkowski sums of large numbers of polytopes

It is known that in the Minkowski sum of r polytopes in dimension d, with r < d, the number of vertices of the sum can potentially be as high as the product of the number of vertices in each summand [2]. However, the number of vertices for sums of more polytopes was unknown so far. In this paper, we study sums of polytopes in general orientations, and show a linear relation between the number of faces of a sum of r polytopes in dimension d, with r ≥ d, and the number of faces in the sums of less than d of the summand polytopes. We deduce from this exact formula a tight bound on the maximum possible number of vertices of the Minkowski sum of any number of polytopes in any dimension. In particular, the linear relation implies that a sum of r polytopes in dimension d has a number of vertices in O(n) of the total number of vertices in the summands, even when r ≥ d. This bound is tight, in the sense that some sums do have that many vertices. ∗Department of Mathematics and Statistics, McGill, Montréal, Canada. [email protected]