Gábor Czédli - Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere ۱, H۶۷۲۰ Hungary
Árpád Kurusa - Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere ۱, Hungary H۶۷۲۰

K. Adaricheva and M. Bolat have recently proved that if \,\mathcal U_۰ and \,\mathcal U_۱ are circles in a triangle with vertices A_۰,A_۱,A_۲, then there exist j\in \{۰,۱,۲\} and k\in\{۰,۱\} such that \,\mathcal U_{۱-k} is included in the convex hull of \,\mathcal U_k\cup(\{A_۰,A_۱, A_۲\}\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_۰ and \,\mathcal  U_۱ are compact sets in the plane such that \,\mathcal U_۱ is obtained from \,\mathcal U_۰ 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.

Congruence lattice, planar semimodular lattice, convex hull, compact set, linebreak circle, combinatorial geometry, abstract convex geometry, anti-exchange property