A convex combinatorial property of compact sets in the plane and its roots in lattice theory
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 inst...
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| Dokumentumtípus: | Cikk |
| Megjelent: |
2019
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| Sorozat: | CATEGORIES AND GENERAL ALGEBRAIC STRUCTURES WITH APPLICATIONS
11 No. Special Issue |
| mtmt: | 30359280 |
| Online Access: | http://publicatio.bibl.u-szeged.hu/15942 |
| Tartalmi kivonat: | 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. |
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| Terjedelem/Fizikai jellemzők: | 57-92 |
| ISSN: | 2345-5853 |