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  "body": "\n## Counterexample\n\nBorsuk's conjecture concerns bounded sets $S \\subseteq \\R^n$ with more than one point, for $n \\ge 1$.\nIt says that every such $S$ can be split into $n + 1$ sets, each of smaller [[diameter|0KW3PN547K]] than $S$.\nThe conjecture is false.\nKahn and Kalai proved this in 1993.\nFor all large $n$, they found finite sets in $\\R^n$ that cannot be split into fewer than $1.2^{\\sqrt{n}}$ parts of smaller diameter.\nThis number exceeds $n + 1$ for large $n$.\n\nTheir sets consist of the matrices $x x^T$, for $x$ in a suitable family of vectors in $\\set{-1, 1}^m$.\nThese matrices are viewed as points of $\\R^{m \\times m}$ with the Euclidean norm.\nFor two such vectors, $\\norm{x x^T - y y^T}^2 = 2 m^2 - 2 (x \\cdot y)^2$.\nSo the diameter is attained exactly by the pairs with $x \\cdot y = 0$, and a part of smaller diameter contains no orthogonal pair.\nThe Frankl–Wilson theorem shows that such parts are exponentially small in $m$.\nSo exponentially many parts are needed, while the dimension is only of order $m^2$.\n\n## Remarks\n\nThe conjecture is true for $n = 2$ and $n = 3$, and for smooth convex bodies in every dimension.\nThe number $n + 1$ cannot be lowered: by the [[Borsuk–Ulam theorem|QHJ0GBQE2E]], the unit ball of $\\R^n$ cannot be split into $n$ closed sets of smaller diameter.\nLater work lowered the dimension of counterexamples; the smallest dimension known to admit a counterexample is $64$, found by Jenrich and Brouwer.\nThe vector $x x^T$ is the [[tensor product|0BF2DE2SYK]] $x \\otimes x$, and the formula above uses $\\norm{x \\otimes x}^2 = \\abs{x}^4 = m^2$ and $\\langle x \\otimes x, y \\otimes y \\rangle = (x \\cdot y)^2$.\n\n## Explanations\n\n### L1 human\n\nBorsuk asked whether any shape in $n$-dimensional space can be cut into $n + 1$ pieces that are all strictly smaller across.\nIn the plane a disc can be cut into three such pieces, and three is needed.\nFor a long time people believed this in every dimension.\nIn 1993 Kahn and Kalai found strange high-dimensional point sets where far more pieces are needed, so the conjecture fails.\n",
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