Circles and crossing planar compact convex sets

Abstract

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.

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

“…Motivated by the proof of (2.8), a more restrictive concept of crossing was introduced in Czédli [18]; it is based on properties of common supporting lines but we will not define it here. Replacing Fejes-Tóth crossing with " [18]-crossing", (2.9) turns into a stronger statement.…”

“…Motivated by the proof of (2.8), a more restrictive concept of crossing was introduced in Czédli [18]; it is based on properties of common supporting lines but we will not define it here. Replacing Fejes-Tóth crossing with " [18]-crossing", (2.9) turns into a stronger statement. Finally, to conclude our mini-survey from George Grätzer's congruence lattices to geometry via a sequence of closely connected consecutive results, we note that Paul Erdős and E. G. Straus [32] extended (2.9) to an analogous characterization of balls in higher dimensions, but the " [18]-crossing" seems to work only in the plane R 2 .…”