Circles and crossing planar compact convex sets

TL;DR

This paper explores various definitions of crossing for convex sets in the plane, establishing a hierarchy among them, and extends Fejes-Tóth's 1967 theorem characterizing disks through these crossing concepts.

Contribution

It introduces a hierarchy of crossing concepts for convex sets, showing that the classical crossing is the least restrictive, and connects geometric crossing notions to combinatorics and lattice theory.

Findings

Each variant of the new crossing concept is more restrictive than the old one.

Fejes-Tóth's theorem holds under the new, more restrictive crossing definitions.

The paper links geometric crossing concepts to combinatorics and lattice theory.

Abstract

Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that whenever $K_1\subseteq \mathbb R^2$ is congruent to $K_0$, then $K_0$ and $K_1$ are not crossing in a natural sense due to L. Fejes-T\'oth. A theorem of L. Fejes-T\'oth 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-T\'oth'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-T\'oth'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.