Cubillages on cyclic zonotopes, membranes, and higher separatation

TL;DR

This paper explores the structural properties of cubillages on cyclic zonotopes and their connections to separated set collections, extending known concepts from lower dimensions to arbitrary dimensions.

Contribution

It introduces new insights into the structure of fine zonotopal tilings on cyclic zonotopes and their relation to higher separated collections.

Findings

Establishes links between cubillages and higher separated collections

Generalizes known results from 2D and 3D to arbitrary dimensions

Provides structural properties of zonotopal tilings

Abstract

We study certain structural properties of fine zonotopal tilings, or cubillages, on cyclic zonotopes $Z(n,d)$ of an arbitrary dimension $d$ and their relations to $(d-1)$-separated collections of subsets of a set $\{1,2,\ldots,n\}$. (Collections of this sort are well known as strongly separated ones when $d=2$, and as chord separated ones when $d=3$.)