Realizing convex codes with axis-parallel boxes
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866910637947355136 |
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| author | Benitez, Miguel Chen, Siran Han, Tianhui Jeffs, R. Amzi Paguyo, Kinapal Zhou, Kevin A. |
| author_facet | Benitez, Miguel Chen, Siran Han, Tianhui Jeffs, R. Amzi Paguyo, Kinapal Zhou, Kevin A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_02486 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Realizing convex codes with axis-parallel boxes Benitez, Miguel Chen, Siran Han, Tianhui Jeffs, R. Amzi Paguyo, Kinapal Zhou, Kevin A. Combinatorics 52C99, 52A20 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. |
| title | Realizing convex codes with axis-parallel boxes |
| topic | Combinatorics 52C99, 52A20 |
| url | https://arxiv.org/abs/2209.02486 |