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Convex
geometries representable by at most five circles on the
plane
Kira Adaricheva, Madina Bolat, Evan Daisy, Ayush Garg, Grace Ma, Michelle Olson, Rohit Pai, Catherine Raanes, Sean Riedel, Joseph Rogge, Raviv S. Sarch, James Thompson, Fernanda Yepez-Lopez and Stephanie Zhou |
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Vol. 17 (2024), No. 2, 337–354
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Abstract |
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A convex geometry is a closure system satisfying the antiexchange property. We document all convex geometries on 4- and 5-element base sets with respect to their representation by circles on the plane. All 34 nonisomorphic geometries on a 4-element set can be represented by circles, and of 672 known geometries on a 5-element set, we give representations for 623. Of the 49 remaining geometries on a 5-element set, one was already shown not to be representable due to the weak carousel property, as articulated by Adaricheva and Bolat (Discrete Math. 342:3 (2019), 726–746). We show that seven more of these convex geometries cannot be represented by circles on the plane, due to what we term the triangular implications property. |
Keywords
convex geometry, convex hull operator for circles,
representation by circles
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Mathematical Subject Classification
Primary: 05B25
Secondary: 06A15
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Supplementary material |
Milestones
Received: 31 May 2023
Revised: 21 January 2024
Accepted: 25 January 2024
Published: 20 May 2024
Communicated by Glenn Hurlbert
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Authors |
| Kira Adaricheva | |
| Hofstra University Hempstead, NY United States |
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| Madina Bolat | |
| University of Illinois Urbana, IL United States |
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| Evan Daisy | |
| Amherst College Amherst, MA United States |
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| Ayush Garg | |
| Indian Institute of Technology New Delhi India |
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| Grace Ma | |
| University of Notre Dame South Bend, IN United States |
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| Michelle Olson | |
| California State University Fullerton, CA United States |
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| Rohit Pai | |
| Georgia Institute of
Technology Atlanta, GA United States |
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| Catherine Raanes | |
| Carnegie Mellon University Pittsburgh, PA United States |
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| Sean Riedel | |
| University of California Santa Cruz, CA United States |
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| Joseph Rogge | |
| University of Washington Seattle, WA United States |
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| Raviv S. Sarch | |
| University of Michigan Ann Arbor, MI United States |
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| James Thompson | |
| University of North Carolina Chapel Hill, NC United States |
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| Fernanda Yepez-Lopez | |
| Indiana University Bloomington, IN United States |
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| Stephanie Zhou | |
| Rutgers University New Brunswick, NJ United States |
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| © 2024 MSP (Mathematical Sciences Publishers). |
