Topological obstructions for vertex numbers of Minkowski sums

We show that for polytopes P₁,P₂,…,Pr⊂Rd, each having ni?d+1 vertices, the Minkowski sum P₁+P₂+?+Pr cannot achieve the maximum of i_∏ni vertices if r?d. This complements a recent result of Fukuda and Weibel (2006), who show that this is possible for up to d−1 summands. The result is obtained by combining methods from discrete geometry (Gale transforms) and topological combinatorics (van Kampen-type obstructions).