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_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.

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

Congruence lattice , planar semimodular lattice , convex hull , compact set , linebreak circle , combinatorial geometry , abstract convex geometry , anti-exchange property