Belt diameter of $\Pi$-zonotopes

TL;DR

This paper establishes upper bounds on the belt diameter of certain zonotopes derived from permutahedra, showing these bounds are tight and relating them to the edge graph diameter of their dual polytopes.

Contribution

It proves new bounds on belt diameters of permutahedron-derived zonotopes and demonstrates their sharpness, linking these to dual polytope edge graph diameters.

Findings

Belt diameter of these zonotopes is at most 3 in general.

For dimensions up to 6, the belt diameter is at most 2.

Bounds on belt diameter are proven to be sharp.

Abstract

We prove that any d-dimensional zonotope obtained from permutahedron by deleting zone vectors has belt diameter at most 3. Moreover if d 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 then 4 and 3 respectively.