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  "body": "\n## Problem\n\nA *universal cover* is a [[convex set|02AS45T3H9]] $C$ in the plane with this property.\nEvery plane set of [[diameter|0KW3PN547K]] $1$ can be moved into $C$ by an [[isometry|02HPP5VCZH]] of the plane.\nFind the smallest possible area of a universal cover.\n\n## Remarks\n\nThe problem is open: the exact minimal area is not known.\nPál showed that the regular hexagon whose opposite sides are at distance $1$ is a universal cover; its area is $\\sqrt{3}/2 \\approx 0.866$.\nCutting suitable pieces off this hexagon gives smaller universal covers, and the best known upper bound is about $0.8441$.\nA disc of diameter $1$ is not a universal cover: the equilateral triangle of side $1$ has diameter $1$ but circumradius $1/\\sqrt{3} \u003e 1/2$.\nLower bounds come from the fact that a universal cover must contain copies of several specific sets of diameter $1$ at once.\nExamples are a disc of diameter $1$ and an equilateral triangle of side $1$.\nThe best known lower bound is about $0.832$.\n\n## Explanations\n\n### L1 human\n\nLebesgue asked for the smallest convex shape that can cover every shape of diameter one, after moving and turning it.\nA regular hexagon of width one works, but it is not the smallest.\nMore than a century later the best possible area is still only known to lie between about $0.832$ and $0.8441$.\n",
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      "first_stated": "1914",
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