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arXiv · 1802.00286

Closed sets with the Kakeya property

Marianna Csörnyei·Kornélia Héra·Miklós Laczkovich
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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.

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Marianna Csörnyei, Kornélia Héra, Miklós Laczkovich. 2018-02-01. Closed sets with the Kakeya property. https://doi.org/10.1112/s0025579316000188

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