%0 Journal Article
%T A convex combinatorial property of compact sets in the plane and its roots in lattice theory
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Czédli, Gábor
%A Kurusa, Árpád
%D 2019
%\ 07/01/2019
%V 11
%N Special Issue Dedicated to Prof. George A. Grätzer
%P 57-92
%! A convex combinatorial property of compact sets in the plane and its roots in lattice theory
%K Congruence lattice
%K planar semimodular lattice
%K convex hull
%K compact set
%K linebreak circle
%K combinatorial geometry
%K abstract convex geometry
%K anti-exchange property
%R 10.29252/cgasa.11.1.57
%X 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.
%U https://cgasa.sbu.ac.ir/article_82639_995ede57b706f33c6488407d8fdd492d.pdf