2016/08/23 by Czédli, Gábor, Kincses, János
#52A01 #Combinatorics (math.CO) #FOS: Mathematics #Primary 05B25 #Secondary 06C10
paper · doi:10.48550/arxiv.1608.06550
Finite convex geometries are combinatorial structures. It follows from a recent result of M. Richter and L.G. Rogers that there is an infinite set Trr of planar convex polygons such that Trr with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of Trr to a finite subset in a natural way. An almost-circle of accuracy 1-ε is a differentiable convex simple closed curve S in the plane having an inscribed circle of radius r1>0 and a circumscribed circle of radius r2 such that the ratio r1/r2 is at least 1-ε. % Motivated by Richter and Rogers' result, we construct a set Tnew such that (1) Tnew contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) Tnew with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to Trr, Tnew is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive ε∈\real and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets E of Tnew such that each E consists of almost-circles of accuracy 1-ε and the convex geometry in question is represented by restricting the convex hull operator to E. The affine-disjointness of E1 and E2 means that, in addition to E1∩ E2=∅, even ψ(E1) is disjoint from E2 for every non-degenerate affine transformation ψ.