Topological obstructions for vertex numbers of Minkowski sums

الأصل · EN

We show that for polytopes P₁, P₂,..., Pᵣ ⊂ ᵈ, 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).