Topological obstructions for vertex numbers of Minkowski sums

Details Topological obstructions for vertex numbers of Minkowski sums Topological obstructions for vertex numbers of Minkowski sums

2007-02-23

arxiv.org/abs/math/0702717v2

Mathematics

Combinatorics

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

Scientific paper

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 R\"{o}rig, Sanyal, and Ziegler (2007).

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