Very recently Richter and Rogers proved that any convex geometry can be represented by a family of convex polygons in the plane. We shall generalize their construction and obtain a wide variety of convex shapes for representing convex geometries. We present an Erdős-Szekeres type obstruction, which answers a question of Czédli negatively, that is general convex geometries cannot be represented with ellipses in the plane. Moreover, we shall prove that one cannot even bound the number of common supporting lines of the pairs of the representing convex sets. In higher dimensions we prove that all convex geometries can be represented with ellipsoids.

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))}.

Abstract. In this paper we discuss the following question: how can we decide whether a convex set is determined by its angle function or not? We give sufficient conditions for convex polygons and for regular convex sets which guarantee that the set is distinguishable. We also investigate the question: which sets are typical (in the sense of Baire category), those which are distinguishable or those which are not? We prove that the family of distinguishable sets is of second Baire category.

No abstract

No abstract