@article {
author = {Czédli, Gábor and Kurusa, Árpád},
title = {A convex combinatorial property of compact sets in the plane and its roots in lattice theory},
journal = {Categories and General Algebraic Structures with Applications},
volume = {11},
number = {Special Issue Dedicated to Prof. George A. Grätzer},
pages = {57-92},
year = {2019},
publisher = {Shahid Beheshti University},
issn = {2345-5853},
eissn = {2345-5861},
doi = {10.29252/cgasa.11.1.57},
abstract = {K. Adaricheva and M. Bolat have recently proved that if $\,\mathcal U_0$ and $\,\mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $\,\mathcal U_{1-k}$ is included in the convex hull of $\,\mathcal U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ for the more general case where $\,\mathcal U_0$ and $\,\mathcal U_1$ are compact sets in the plane such that $\,\mathcal U_1$ is obtained from $\,\mathcal U_0$ by a positive homothety or by a translation. Also, we give a short survey to show how lattice theoretical antecedents, including a series of papers on planar semimodular lattices by G. Grätzer and E. Knapp, lead to our result.},
keywords = {Congruence lattice,planar semimodular lattice,convex hull,compact set,linebreak circle,combinatorial geometry,abstract convex geometry,anti-exchange property},
url = {https://cgasa.sbu.ac.ir/article_82639.html},
eprint = {https://cgasa.sbu.ac.ir/article_82639_995ede57b706f33c6488407d8fdd492d.pdf}
}