Representation of convex geometries by circles on a plane

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.

Section: Preliminariesmentioning

confidence: 95%

Section: Erdős-szekeres Type Obstructionsmentioning

confidence: 99%

Section: Ellipsoids In Higher Dimensionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the representation of finite convex geometries with convex sets

Kincses¹

2017

Characterizing circles by a convex combinatorial property

Czédli¹

2017

Abstract. Let K 0 be a compact convex subset of the plane R 2 , and assume that K 1 ⊆ R 2 is similar to K 0 , that is, K 1 is the image of K 0 with respect to a similarity transformation R 2 → R 2 . Kira Adaricheva and Madina Bolat have recently proved that if K 0 is a disk and both K 0 and K 1 are included in a triangle with vertices A 0 , A 1 , and A 2 , then there exist a j ∈ {0, 1, 2} and a k ∈ {0, 1} such thatHere we prove that this property characterizes disks among compact convex subsets of the plane. In fact, we prove even more since we replace "similar" by "isometric" (also called "congruent"). Circles are the boundaries of disks, so our result also gives a characterization of circles. Aim and introductionOur goal. The real plane and the usual convex hull operator on it will be denoted by R 2 and Conv, respectively. That is, for a set X ⊆ R 2 of points, Conv(X) is the smallest convex subset of R 2 that contains X. As usual, if X and Y are subsets of R 2 such that Y = ϕ(X) for a similarity transformation or an isometric transformation ϕ : R 2 → R 2 , then X and Y are similar or isometric (also called congruent), respectively. Disks are convex hulls of circles and circles are boundaries of disks. The singleton subsets of R 2 are both disks and circles. A compact subset of R 2 is a topologically closed and bounded subset. Our aim is to prove the following theorem. Theorem 1.1. If K 0 is a compact convex subset of the plane R 2 , then the following three conditions are equivalent.(i) K 0 is a disk.(ii) For every K 1 ⊆ R 2 and for arbitrary points A 0 , A 1 , A 2 ∈ R 2 , if K 1 is similar to K 0 and both K 0 and K 1 are contained in the triangle Conv({A 0 , A 1 , A 2 }), then there exist a j ∈ {0, 1, 2} and a k ∈ {0, 1} such that(iii) The same as the second condition but "similar" is replaced by "isometric".Our main achievement is that (iii) implies (i). The implication (i) ⇒ (ii) was discovered and proved by Adaricheva and Bolat [3]; see also Czédli [9] for a shorter proof. The implication (ii) ⇒ (iii) is trivial.

Circles and crossing planar compact convex sets

Czédli¹

2019

Let K 0 be a compact convex subset of the plane R 2 , and assume that whenever K 1 ⊆ R 2 is congruent to K 0 , then K 0 and K 1 are not crossing in a natural sense due to L. Fejes-Tóth. A theorem of L. Fejes-Tóth from 1967 states that the assumption above holds for K 0 if and only if K 0 is a disk. In a paper appeared in 2017, the present author introduced a new concept of crossing, and proved that L. Fejes-Tóth's theorem remains true if the old concept is replaced by the new one. Our purpose is to describe the hierarchy among several variants of the new concepts and the old concept of crossing. In particular, we prove that each variant of the new concept of crossing is more restrictive then the old one. Therefore, L. Fejes-Tóth's theorem from 1967 becomes an immediate consequence of the 2017 characterization of circles but not conversely. Finally, a mini-survey shows that this purely geometric paper has precursor in combinatorics and, mainly, in lattice theory.