Maximal f-Vectors of Minkowski Sums of Large Numbers of Polytopes

Abstract

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 be as high as the product of the number of vertices in each summand. However, the number of vertices for sums of more polytopes was unknown so far. In this paper, we study sums of polytopes and prove 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 result a 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, where summands have n vertices in total, has less than ${n \choose{d-1}}$ vertices, even when r≥d. Finally, we present a construction for any given number of vertices in summands and show that no other sum can achieve more vertices, establishing a precise tight bound.