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

Czédli, Gábor and Kurusa, Árpád (2019) A convex combinatorial property of compact sets in the plane and its roots in lattice theory. CATEGORIES AND GENERAL ALGEBRAIC STRUCTURES WITH APPLICATIONS, 11 (Specia). pp. 57-92. ISSN 2345-5853

Preview

Text 1807.03443.pdf Download (662kB) | Preview

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. Gr\"atzer and E. Knapp, lead to our result.

Actions (login required)

Edit Item