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CLOSED SETS WITH THE KAKEYA PROPERTY
Published online by Cambridge University Press: 23 September 2016
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 non-trivial 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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- Research Article
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- Copyright © University College London 2016
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