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

2018/07/10 by Czédli, Gábor, Kurusa, Árpád · 1 citation
#06C10 (secondary) #52A01 #52C99 (Primary) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1807.03443

Abstract

K. Adaricheva and M. Bolat have recently proved that if U₀ and U₁ are circles in a triangle with vertices A₀,A₁,A₂, then there exist j∈ \0,1,2\ and k∈\0,1\ such that U_1-k is included in the convex hull of U_k∪(\A₀,A₁, A₂\∖\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 U₀ and U₁ are compact sets in the plane such that U₁ is obtained from U₀ 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.

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• Infinite convex geometries with lower semi-modularity and join semi-distributivity

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