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Kakeya set
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==Kakeya needle problem== The '''Kakeya needle problem''' asks whether there is a minimum area of a region <math>D</math> in the plane, in which a needle of unit length can be turned through 360°. This question was first posed, for [[Convex set|convex]] regions, by {{harvs|txt|author-link=Sōichi Kakeya|first=Sōichi|last= Kakeya|year=1917}}. The minimum area for convex sets is achieved by an [[equilateral triangle]] of height 1 and area 1/{{radic|3}}, as [[Gyula Pál|Pál]] showed.<ref>{{cite journal| last = Pal | first = Julius | title = Ueber ein elementares variationsproblem | journal = Kongelige Danske Videnskabernes Selskab Math.-Fys. Medd. | volume = 2 | pages = 1–35 | year = 1920 }}</ref> Kakeya seems to have suggested that the Kakeya set <math>D</math> of minimum area, without the convexity restriction, would be a three-pointed [[deltoid curve|deltoid]] shape. However, this is false; there are smaller non-convex Kakeya sets.
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