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 U0 and U1 are circles in a triangle with vertices A0, A1, A2, then there exist j ∈ {0, 1, 2} and k ∈ {0, 1} such that U1−k is included in the convex hull of U k ∪ ({A0, A1, A2} \ {Aj}). One could say disks instead of circles. Here we prove the existence of such a j and k for the more general case where U0 and U1 are compact sets in the plane such that U1 is obtained from U0 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.