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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 |
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Vol. 17 (2024), No. 4, 633–649
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Abstract |
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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 -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
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Mathematical Subject Classification
Primary: 52A20, 52C99
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Milestones
Received: 20 September 2022
Revised: 31 May 2023
Accepted: 5 June 2023
Published: 2 October 2024
Communicated by Steven J. Miller
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Authors |
| Miguel Benitez | |
| Carnegie Mellon University Pittsburgh, PA United States |
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| Siran Chen | |
| Carnegie Mellon University Pittsburgh, PA United States |
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| Tianhui Han | |
| Carnegie Mellon University Pittsburgh, PA United States |
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| R. Amzi Jeffs | |
| Carnegie Mellon University Pittsburgh, PA United States |
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| Kinapal Paguyo | |
| Carnegie Mellon University Pittsburgh, PA United States |
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| Kevin A. Zhou | |
| Carnegie Mellon University Pittsburgh, PA United States |
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| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). |
