Belt distance between facets of space-filling zonotopes

TL;DR

This paper establishes an upper bound on the belt diameter of space-filling zonotopes in any dimension, showing it grows logarithmically with dimension and is tight in low dimensions.

Contribution

It proves a new logarithmic upper bound on belt diameter for space-filling zonotopes and demonstrates the bound's optimality in dimensions up to six.

Findings

Belt diameter of space-filling zonotopes is at most rac{rac{4}{5}d}

The bound is tight for dimensions up to 6

Provides a geometric interpretation via a 'subway map' analogy

Abstract

For every d-dimensional polytope P with centrally symmetric facets we can associate a "subway map" such that every line of this "subway" corresponds to set of facets parallel to one of ridges P. The belt diameter of P is the maximal number of line changes that you need to do in order to get from one station to another. In this paper we prove that belt diameter of d-dimensional space-filling zonotope is not greater than $\lceil \log_2\frac45d\rceil$. Moreover we show that this bound can not be improved in dimensions d at most 6.