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

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    K. Adaricheva and M. Bolat have recently proved that if U0U_0 and U1U_1 are circles in a triangle with vertices A0,A1,A2A_0,A_1,A_2, then there exist j{0,1,2}j\in \{0,1,2\} and k{0,1}k\in\{0,1\} such that U1kU_{1-k} is included in the convex hull of Uk({A0,A1,A2}{Aj})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 jj and kk for the more general case where U0U_0 and U1U_1 are compact sets in the plane such that U1U_1 is obtained from U0U_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.Comment: 28 pages, 7 figure
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