Topological obstructions for vertex numbers of Minkowski sums

We show that for polytopes P1, P2, . . . , Pr ⊂ R, 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 & 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) as developed in Rörig, Sanyal, and Ziegler (2007).