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

Abstract

K. Adaricheva and M. Bolat have recently proved that if U 0 and U 1 are circles in a triangle with vertices A0, A1, A2, then there exist j ∈ {0, 1, 2} and k ∈ {0, 1} such that U 1−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 U 0 and U 1 are compact sets in the plane such that U 1 is obtained from 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.

Revisiting Faigle geometries from a perspective of semimodular lattices

“…These lattices are finite and necessarily planar. Four dozen papers (including the present one) have been devoted to these lattices and their applications since then; see Czédli and Grätzer [4], Czédli and Kurusa [5], the "mini-survey" subsection of Czédli [3], and their references for most of these four dozen papers. For a full list, see www.math.u-szeged.hu/~czedli/m/listak/publ-psml.pdf.…”