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Kakeya set
In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensional space, forms a Kakeya set. Much of the research in this area has studied the problem of how small such sets can be. Besicovitch showed that there are Besicovitch sets of measure zero.
A Kakeya needle set (sometimes also known as a Kakeya set) is a (Besicovitch) set in the plane with a stronger property, that a unit line segment can be rotated continuously through 180 degrees within it, returning to its original position with reversed orientation. Again, the disk of radius 1/2 is an example of a Kakeya needle set.
==Kakeya needle problem==
The Kakeya needle problem asks whether there is a minimum area of a region ''D'' in the plane, in which a needle of unit length can be turned through 360°. This question was first posed, for convex regions, by . The minimum area for convex sets is achieved by an equilateral triangle of height 1 and area 1/√3, as Pál showed.
Kakeya seems to have suggested that the Kakeya set ''D'' of minimum area, without the convexity restriction, would be a three-pointed deltoid shape. However, this is false; there are smaller non-convex Kakeya sets.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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