Topological obstructions for vertex numbers of Minkowski sums

Abstract

We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i \ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the maximum of \prod_i n_i vertices if r \ge 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).

13 pages, 2 figures; Improved exposition and less typos. Construction/example and remarks added

References in corpus (2)

• On f-vectors of Minkowski additions of convex polytopes
• Non-projectability of polytope skeleta

Cited by in corpus (7)

• Construction and Analysis of Projected Deformed Products
• Non-projectability of polytope skeleta
• Prodsimplicial-Neighborly Polytopes
• Multitriangulations, pseudotriangulations and some problems of realization of polytopes
• Tight lower bounds on the number of faces of the Minkowski sum of convex polytopes via the Cayley trick
• The convex dimension of hypergraphs and the hypersimplicial Van Kampen-Flores Theorem
• Maximal f-vectors of Minkowski sums of large numbers of polytopes