Topological obstructions for vertex numbers of Minkowski sums
Combinatorics
2012-12-27 v2 Metric Geometry
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 R\"{o}rig, Sanyal, and Ziegler (2007).
Keywords
Cite
@article{arxiv.math/0702717,
title = {Topological obstructions for vertex numbers of Minkowski sums},
author = {Raman Sanyal},
journal= {arXiv preprint arXiv:math/0702717},
year = {2012}
}
Comments
13 pages, 2 figures; Improved exposition and less typos. Construction/example and remarks added