Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701A convex combinatorial property of compact sets in the plane and its roots in lattice theory579282639ENGábor CzédliBolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, H6720 Hungary0000-0001-9990-3573Árpád KurusaBolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, Hungary H6720Journal Article20180709K. 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 $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.<br />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. <br />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.http://cgasa.sbu.ac.ir/article_82639_995ede57b706f33c6488407d8fdd492d.pdf