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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

Vol. 17 (2024), No. 2, 337–354
Abstract

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
Mathematical Subject Classification
Primary: 05B25
Secondary: 06A15
Supplementary material

Appendix A: Descriptions of geometries of sizes 4 and 5

Appendix B: Representations of geometries of sizes 4 and 5

Milestones
Received: 31 May 2023
Revised: 21 January 2024
Accepted: 25 January 2024
Published: 20 May 2024

Communicated by Glenn Hurlbert
Authors
Kira Adaricheva
Hofstra University
Hempstead, NY
United States
Madina Bolat
University of Illinois
Urbana, IL
United States
Evan Daisy
Amherst College
Amherst, MA
United States
Ayush Garg
Indian Institute of Technology
New Delhi
India
Grace Ma
University of Notre Dame
South Bend, IN
United States
Michelle Olson
California State University
Fullerton, CA
United States
Rohit Pai
Georgia Institute of Technology
Atlanta, GA
United States
Catherine Raanes
Carnegie Mellon University
Pittsburgh, PA
United States
Sean Riedel
University of California
Santa Cruz, CA
United States
Joseph Rogge
University of Washington
Seattle, WA
United States
Raviv S. Sarch
University of Michigan
Ann Arbor, MI
United States
James Thompson
University of North Carolina
Chapel Hill, NC
United States
Fernanda Yepez-Lopez
Indiana University
Bloomington, IN
United States
Stephanie Zhou
Rutgers University
New Brunswick, NJ
United States