On the representation of finite convex geometries with convex sets

Kincses, János

doi:10.14232/actasm-017-502-z

Cited by 9 publications

(9 citation statements)

References 15 publications

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“…In a forthcoming paper, we will use Lemma 3.5 to establish a connection between the present paper and Fejes-Tóth [13]. Finally, we note that our topic is also in connection with a quite recent paper by Kincses [15].…”

Section: Figure 12 No "Tangent Interval" Is Possiblementioning

confidence: 68%

Characterizing circles by a convex combinatorial property

Czédli¹

2017

Acta Sci. Math.

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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.

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Section: Figure 12 No "Tangent Interval" Is Possiblementioning

confidence: 68%

Characterizing circles by a convex combinatorial property

Czédli¹

2017

Acta Sci. Math.

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“…This is such an obstacle that does not allow to represent every convex geometry by circles. Even more is true; later, Kincses [53] found an Erdős-Szekeres type obstruction for representing convex geometries by ellipses. Similarly to ellipses, he could exclude many other shapes.…”

Section: Some Results Of Geometrical Naturementioning

confidence: 99%

“…Kincses [53] proved that every convex geometry can be represented by ellipsoids in R n for some n ∈ N + := {1, 2, 3, . .…”

Section: Some Results Of Geometrical Naturementioning

confidence: 99%

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

Czédli

Kurusa

2019

CGASA

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K. Adaricheva and M. Bolat have recently proved that if U 0 and U 1 are circles in a triangle with vertices A0, A1, A2, then there exist j ∈ {0, 1, 2} and k ∈ {0, 1} such that U 1−k is included in the convex hull of U k ∪ ({A0, A1, A2} \ {Aj}). One could say disks instead of circles. Here we prove the existence of such a j and k for the more general case where U 0 and U 1 are compact sets in the plane such that U 1 is obtained from U 0 by a positive homothety or by a translation. Also, we give a short survey to show how lattice theoretical antecedents, including a series of papers on planar semimodular lattices by G. Grätzer and E. Knapp, lead to our result.

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“…The question whether ellipses rather than circles are appropriate to represent all finite convex geometries was raised in Czédli [11]. This question has recently been answered in negative by Kincses [33], who presented an Erdős-Szekeres type obstruction to such a representation.…”

Section: From Congruence Lattices To the Present Papermentioning

confidence: 99%

Circles and crossing planar compact convex sets

Czédli¹

2019

Acta Sci. Math.

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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.

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On the representation of finite convex geometries with convex sets

Cited by 9 publications

References 15 publications

Characterizing circles by a convex combinatorial property

Characterizing circles by a convex combinatorial property

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

Circles and crossing planar compact convex sets

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