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

Gábor Czédli; Árpád Kurusa

paper

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

arXiv:1807.03443

Abstract

K. Adaricheva and M. Bolat have recently proved that if $U_{0}$ and $U_{1}$ 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 $U_{1 - k}$ is included in the convex hull of $U_{k} \cup ({A_{0}, A_{1}, A_{2}} ∖ {A_{j}})$ . One could say disks instead of circles. Here we prove the existence of such a $j$ and $k$ for the more general case where $U_{0}$ 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. Gratzer and E. Knapp, lead to our result.

arXiv comment28 pages, 7 figures