A convex combinatorial property of compact sets in the plane and its roots in lattice theory

K. Adaricheva and M. Bolat have recently proved that if \,\mathcal U_۰ and \,\mathcal U_۱ are circles in a triangle with vertices A_۰,A_۱,A_۲, then there exist j\in \{۰,۱,۲\} and k\in\{۰,۱\} such that \,\mathcal U_{۱-k} is included in the convex hull of \,\mathcal U_k\cup(\{A_۰,A_۱, A_۲\}\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_۰ and \,\mathcal  U_۱ are compact sets in the plane such that \,\mathcal U_۱ is obtained from \,\mathcal U_۰ 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