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  "body": "\n## Description\n\nLet $n \\ge 2$.\nA *Kakeya set*, or *Besicovitch set*, is a [[compact set|0R6EWFKRCF]] $K \\subseteq \\R^n$ that contains a line segment of length $1$ in every direction.\nThat is, for each unit vector $e \\in \\R^n$ there is a point $a \\in \\R^n$ with $\\set{a + t e : 0 \\le t \\le 1} \\subseteq K$.\nThe closed ball of radius $1/2$ about the origin is a Kakeya set.\nFor a unit vector $e$, the segment from $-e/2$ to $e/2$ lies in the ball.\nBesicovitch showed that there are Kakeya sets in $\\R^2$ with [[Lebesgue measure|2Z2QVWZ262]] zero.\nIf $K \\subseteq \\R^2$ is such a set, then $K \\times [0,1]^{n-2}$ is a Kakeya set in $\\R^n$ of measure zero.\nIndeed, let $e = (e_1, e_2, e')$ be a unit vector.\nTake a segment of $K$ in the direction of $(e_1, e_2)$ with length $\\abs{(e_1, e_2)}$.\nThe last $n - 2$ coordinates then move by $te'$ for $0 \\le t \\le 1$, starting from a suitable corner of the cube.\n\n## Remarks\n\nAlthough a Kakeya set may have measure zero, Davies proved in 1971 that every Kakeya set in $\\R^2$ has [[Hausdorff dimension|SSNFTHZ6YJ]] $2$.\nThe Kakeya conjecture asks whether every Kakeya set in $\\R^n$ has Hausdorff dimension $n$.\nIn 2025 Hong Wang and Joshua Zahl posted a proof for $n = 3$.\nSome authors do not require a Kakeya set to be compact.\n\n## Explanations\n\n### L1 human\n\nA Kakeya set is a set in which a needle of length one can be found pointing in every direction.\nA disc of diameter one works, since a diameter can point in any direction.\nThe surprise is that such a set can be so thin that it has area zero.\nIts needles overlap in a very clever way, so that very little room is used.\n",
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