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

TL;DR

This paper extends a geometric property involving circles in a triangle to more general compact sets in the plane, linking it to lattice theory and demonstrating the existence of specific inclusion relations.

Contribution

It generalizes a known convex combinatorial property from circles to compact sets related by homothety or translation, and connects the result to lattice theory origins.

Findings

Existence of a specific inclusion relation for compact sets under homothety or translation.

Connection between geometric properties and lattice theoretical structures.

Survey of lattice theory's influence on geometric convexity results.

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 $U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. One could say disks instead of circles. Here we prove the existence of such a $j$ and $k$ for the more general case where $U_0$ 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.