Circles and crossing planar compact convex sets

Czédli, Gábor: Circles and crossing planar compact convex sets. In: Acta scientiarum mathematicarum, (85) 1-2. pp. 337-353. (2019)

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Let K0 be a compact convex subset of the plane R 2 , and assume that whenever K1 ⊆ R 2 is congruent to K0, then K0 and K1 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 K0 if and only if K0 is a disk. In a paper that 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 than 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 precursors in combinatorics and, mainly, in lattice theory.

Item Type: Article
Journal or Publication Title: Acta scientiarum mathematicarum
Date: 2019
Volume: 85
Number: 1-2
ISSN: 2064-8316
Page Range: pp. 337-353
Official URL:
Uncontrolled Keywords: Matematika
Additional Information: Bibliogr.: p. 351-353. ; összefoglalás angol nyelven
Date Deposited: 2019. Sep. 25. 11:03
Last Modified: 2019. Sep. 25. 11:03

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