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Belt distance between facets of space-filling zonotopes. (English. Russian original) Zbl 1275.52012
Math. Notes 92, No. 3, 345-355 (2012); translation from Mat. Zametki 92, No. 3, 381-394 (2012).
A belt of a polytope \(P\) is the set of all facets parallel to a given \((n-2)\)-face of \(P\). A sequence of facets is called a belt path if every two consecutive facets in the sequence belong to the same belt. The number of different belt the facets in a belt path belong to is called the length of the path. Finally the belt distance of two facets is the length of the shortest belt path between them and accordingly, the belt diameter of a polytope \(P\) is the maximal belt distance between any two facets of \(P\).
The author investigates belt diameters of zonotopes that are also parallelotopes and proves an upper bound of \(\log_2(\frac45 d)\) for \(d\)-dimensional space-filling zonotopes. To this end, he shows that it is enough to consider zonotopes whose generators lie in two conjugate sets and which behave nicely under projections in a certain direction. Afterwards he uses an inductive argument to show the statement for those special zonotopes.
The author further shows that this bound is sharp in dimensions up to \(6\).
MSC:
52B11 \(n\)-dimensional polytopes
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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