Representing convex geometries by almost-circles

Abstract

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 T RR of planar convex polygons such that T RR with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of T RR to a finite subset in a natural way. An almost-circle of accuracy 1 − ǫ is a differentiable convex simple closed curve S in the plane having an inscribed circle of radius r 1 > 0 and a circumscribed circle of radius r 2 such that the ratio r 1 /r 2 is at least 1−ǫ. 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 T RR , Tnew is closed with respect to nondegenerate affine transformations; and (4) for every (small) positive ǫ ∈ R and for every finite convex geometry, there are continuum many pairwise affinedisjoint finite subsets E of Tnew such that each E consists of almost-circles of accuracy 1−ǫ and the convex geometry in question is represented by restricting the convex hull operator to E. The affine-disjointness of E 1 and E 2 means that, in addition to E 1 ∩ E 2 = ∅, even ψ(E 1 ) is disjoint from E 2 for every non-degenerate affine transformation ψ. Date: August 23, 2016. 1991 Mathematics Subject Classification. Primary 05B25; Secondary 06C10, 52A01. Key words and phrases. Abstract convex geometry, anti-exchange system, differentiable curve, almost-circle. This research was supported by NFSR of Hungary (OTKA), grant number K 115518. 1 (1.3) Points(X) = C∈X C, and Conv T (X) : = {D ∈ T : D ⊆ Conv R 2 (Points(X))}.

Circles and crossing planar compact convex sets

“…Recently, various representation theorems are available for convex geometries and for the corresponding lattices; we mention only Adaricheva [1], Adaricheva and Czédli [3], Adaricheva, Gorbunov and Tumanov [4], Adaricheva and Nation [5] and [6], Czédli [10], Czédli and Kincses [17], Kashiwabara, Nakamura, and Okamoto [32], and Richter and Rogers [35]. Czédli [11] gave a lattice theoretical approach to a new sort of representation, in which some convex geometries were represented by circles.…”

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

“…First, Richter and Rogers [56] represented every convex geometry analogously to (2.6) but using polygons instead of circles. Second, Czédli and Kincses [22] replaced polygons with objects taken from an appropriate family of so-called "almost circles". However, it was not known at that time whether circles would do instead of "almost circles".…”

Conditional Logic is Complete for Convexity in the Plane

“…This step is not trivial because, as we already explained above, not every finite convex geometry is isomorphic to a subspace of some Euclidean space. However, following [20], there has recently been a rich literature on representing finite convex geometries inside of Euclidean spaces [10,11,32,2]. The main result of [20], for which [32] give a much shorter proof, is that every finite convex geometry is isomorphic to the convex geometry of a finite set of points in the plane, if we use an alternative notion of convex set that is slightly different from the standard notion of convex set in the plane.…”