Belt diameter of ΠΠ-zonotopes

A ΠΠ-zonotope is a zonotope that can be obtained from permutahedron by deleting zone vectors. Any face FF of codimension 2 of such zonotope generates its belt  , i.e. the set of all facets parallel to FF. The belt diameter   of a given zonotope ZZ is the diameter of the graph with vertices correspondent to pairs of opposite facets and with edges connect facets in one belt.In this paper we investigate belt diameters of ΠΠ-zonotopes. We prove that any dd-dimensional ΠΠ-zonotope (d≥3d≥3) has belt diameter at most 3. Moreover if dd is not greater than 6 then its belt diameter is bounded from above by 2. Also we show that these bounds are sharp. As a consequence we show that diameter of the edge graph of dual polytope for such zonotopes is not greater than 4 and 3 respectively.