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  "body": "\n## Problem\n\nA needle is a line segment of length $1$ in the plane $\\R^2$.\nFind the smallest area, in the sense of [[Lebesgue measure|2Z2QVWZ262]], of a region $R \\subset \\R^2$ with the following property.\nThe needle can be moved continuously inside $R$ so that it ends in its starting position with its two ends exchanged.\n\n## Remarks\n\nThe problem is solved, and the answer is that there is no positive minimum.\nAbram Besicovitch showed that for every $\\varepsilon \u003e 0$ there is a region of area less than $\\varepsilon$ in which the needle can be turned around.\nHis construction builds on a set of measure zero that contains a unit segment in every direction, an example of a [[Kakeya set|6PEN7AKC7M]].\nIf $R$ is required to be convex, the minimum exists: it is the equilateral triangle of height $1$, of area $1/\\sqrt{3}$, a result of Julius Pál.\n\n## Explanations\n\n### L1 human\n\nImagine a stick of length one lying on a table, and you must turn it to face the other way while keeping it inside a painted region.\nA disc of diameter one works, with area about $0.785$, and a triangle of height one does better.\nSurprisingly, with clever back-and-forth moves the painted region can be made as small in area as you like.\n",
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      "first_stated": "1917",
      "posed_by": "Sōichi Kakeya"
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