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

Document Type : Research Paper

Authors

1 Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, H6720 Hungary

2 Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, Hungary H6720

10.29252/cgasa.11.1.57

Abstract

K. Adaricheva and M. Bolat have recently proved that if $\,\mathcal U_0$ and $\,\mathcal U_1$ 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 $\,\mathcal U_{1-k}$ is included in the convex hull of $\,\mathcal U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. One could say disks instead of circles.
Here we prove the existence of such a $j$ and $k$ for the more general case where $\,\mathcal U_0$ and $\,\mathcal  U_1$ are compact sets in the plane such that $\,\mathcal U_1$ is obtained from $\,\mathcal U_0$ 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.

Keywords

References

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