具有Kakeya性质的闭集
Closed sets with the Kakeya property
摘要：
我们说平面集合$A$具有Kakeya性质，如果存在$A$的两个不同位置，使得$A$可以在任意小的区域内从第一个位置连续移动到第二个位置.证明了如果$A$是闭的并且具有Kakeya性质，则$A$的非平凡连通分量的并可以被一个零集覆盖，该空集要么是平行线的并，要么是同心圆的并。特别地，如果$A$是闭合的、连通的并且具有Kakeya性质，则$A$可以被线或圆覆盖。
英文摘要：
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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