Realizing convex codes with axis-parallel boxes

TL;DR

This paper characterizes the combinatorial codes realizable by axis-parallel boxes in Euclidean space, introduces a product theorem relating codes of set collections, and explores differences with convex set codes.

Contribution

It presents a general product theorem for codes of set collections and characterizes codes realizable by axis-parallel boxes, highlighting differences from convex set codes.

Findings

Characterization of codes realizable by axis-parallel boxes

A product theorem relating codes of combined set collections

Differences identified between box and convex set codes

Abstract

Every ordered collection of sets in Euclidean space can be associated to a combinatorial code, which records the regions cut out by the sets in space. Given two ordered collections of sets, one can form a third collection in which the $i$-th set is the Cartesian product of the corresponding sets from the original collections. We prove a general "product theorem" which characterizes the code associated to the collection resulting from this operation, in terms of the codes associated to the original collections. We use this theorem to characterize the codes realizable by axis-parallel boxes, and exhibit differences between this class of codes and those realizable by convex open or closed sets. We also use our theorem to prove that a "monotonicity of open convexity" result of Cruz, Giusti, Itskov, and Kronholm holds for closed sets when some assumptions are slightly weakened.