Topological obstructions for vertex numbers of Minkowski sums

We show that for polytopes P 1 , P 2 , … , P r ⊂ R d , each having n i ⩾ d + 1 vertices, the Minkowski sum P 1 + P 2 + ⋯ + P r cannot achieve the maximum of ∏ i n i 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).