This might be a very simple question, but the answer is not entirely obvious to me and I've been unable to find it stated anywhere. Consider two sets of generators (vectors), which are reduced in the sense that no two generators are parallel to each other (if one starts with two parallel generators, reduce that set by adding them together to make a single one). Then, if these two sets are not identical, can they possibly represent the same zonotope? I believe the answer is no. Am I wrong? A proof one way or the other would be appreciated.
2026-03-26 19:17:12.1774552632
Uniqueness of generator set for zonotope
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As I suspected, this is probably kind of obvious. Since every edge of a given zonotope is a translate of one of its generators (see https://hal.science/hal-03409685/document), then if any generator of the first zonotope is not a generator of the second, the first will have an edge that the second does not. In such a case, the two are obviously not the same zonotope. Thus, if two zonotopes are equivalent, they must have the same set of generators.