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Realizing convex codes with axis-parallel boxes

Miguel Benitez, Siran Chen, Tianhui Han, R. Amzi Jeffs, Kinapal Paguyo and Kevin A. Zhou

Vol. 17 (2024), No. 4, 633–649
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.

Keywords
convex, code, box, realization, axis-parallel
Mathematical Subject Classification
Primary: 52A20, 52C99
Milestones
Received: 20 September 2022
Revised: 31 May 2023
Accepted: 5 June 2023
Published: 2 October 2024

Communicated by Steven J. Miller
Authors
Miguel Benitez
Carnegie Mellon University
Pittsburgh, PA
United States
Siran Chen
Carnegie Mellon University
Pittsburgh, PA
United States
Tianhui Han
Carnegie Mellon University
Pittsburgh, PA
United States
R. Amzi Jeffs
Carnegie Mellon University
Pittsburgh, PA
United States
Kinapal Paguyo
Carnegie Mellon University
Pittsburgh, PA
United States
Kevin A. Zhou
Carnegie Mellon University
Pittsburgh, PA
United States