Closed sets with the Kakeya property

TL;DR

This paper characterizes closed sets with the Kakeya property in the plane, showing they are essentially unions of lines or circles, and provides a classification for connected sets with this property.

Contribution

It proves that closed sets with the Kakeya property are covered by lines or circles, offering a complete geometric classification.

Findings

Closed sets with the Kakeya property are unions of lines or circles.

Connected sets with the Kakeya property are contained in a single line or circle.

The union of nontrivial components of such sets can be covered by null sets of lines or circles.

Abstract

We say that a planar set $A$ has the Kakeya property if there exist two different positions of $A$ such that $A$ can be continuously moved from the first position to the second within a set of arbitrarily small area. We prove that if $A$ is closed and has the Kakeya property, then the union of the nontrivial connected components of $A$ can be covered by a null set which is either the union of parallel lines or the union of concentric circles. In particular, if $A$ is closed, connected and has the Kakeya property, then $A$ can be covered by a line or a circle.