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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 and are
circles in a triangle with vertices , then there exist and such that is included in the convex hull
of . One could say disks instead of
circles. Here we prove the existence of such a and for the more general
case where and are compact sets in the plane such that is
obtained from 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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