Belt distance between facets of space-filling zonotopes

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.Comment: 17 pages, 5 figure

Lattice zonotopes of degree 2

The Ehrhart polynomial $ehr_P (n)$ of a lattice polytope $P$ gives the number of integer lattice points in the $n$-th dilate of $P$ for all integers $n\geq 0$. The degree of $P$ is defined as the degree of its $h^\ast$-polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree $2$ thereby complementing results of Scott (1976), Treutlein (2010), and Henk-Tagami (2009). Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all $3$-dimensional zonotopes of degree $2$.Comment: 12 pages, 1 figure; v2: minor revision

Fluctuations of lattice zonotopes and polygons

Following Barany et al., who proved that large random lattice zonotopes converge to a deterministic shape in any dimension after rescaling, we establish a central limit theorem for finite-dimensional marginals of the boundary of the zonotope. In dimension 2, for large random convex lattice polygons contained in a square, we prove a Donsker-type theorem for the boundary fluctuations, which involves a two-dimensional Brownian bridge and a drift term that we identify as a random cubic curve