Representing convex geometries by almost-circles

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. For a (small) nonnegative ɛ < 1, a differentiable convex simple closed planar curve S will be called an almost-circle of accuracy 1 − ɛ if it lies in an annulus of radii 0 < r1 ≤ r2 such that r1/r2 ≥ 1 − ɛ. 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) Tnew is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive ɛ ∈ R 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 − ɛ 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.