ACTA issues

Representing convex geometries by almost-circles

Gábor Czédli, János Kincses

Acta Sci. Math. (Szeged) 83:3-4(2017), 393-414

Abstract. \emph{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 $\epsilon < 1$, a differentiable convex simple closed planar curve $S$ will be called an \emph{almost-circle of accuracy} $1-\epsilon $ if it lies in an annulus of radii $0< r_1\leq r_2$ such that $r_1/r_2 \geq1-\epsilon $. Motivated by Richter and Rogers' result, we construct a set $\Tczk $ such that (1) $\Tczk $ 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) $\Tczk $ with respect to the geometric convex hull operator is a locally convex geometry; (3) $\Tczk $ is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive $\epsilon\in \real $ and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets $E$ of $\Tczk $ such that each $E$ consists of almost-circles of accuracy $1-\epsilon $ 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\cap E_2=\emptyset $, even $\psi(E_1)$ is disjoint from $E_2$ for every non-degenerate affine transformation $\psi $.

DOI: 10.14232/actasm-016-044-8

AMS Subject Classification (1991): 05B25; 06C10, 52A01

Keyword(s): abstract convex geometry, anti-exchange system, differentiable curve, almost-circle

Received August 23, 2016, and in revised form June 27, 2017. (Registered under 44/2016.)