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  "body": "\n## Statement\n\nLet $n \\ge 1$.\nLet $T$ be a regular simplex in $\\R^n$ with edge length $2$.\nLet $\\sigma_n$ be the fraction of the volume of $T$ covered by the $n + 1$ unit balls centred at the vertices of $T$.\nThen every [[sphere packing|HF32CBN5MA]] of unit balls in $\\R^n$ has upper [[packing density|4P5PSWADJD]] at most $\\sigma_n$.\n\n## Remarks\n\nC. A. Rogers proved the bound in 1958.\nThe proof splits space into the Voronoi cells of the centres, and splits each cell further into simplices.\nA local volume argument shows that the balls fill at most the fraction $\\sigma_n$ of each Voronoi cell.\nFor $n = 2$, the triangle of side $2$ has area $\\sqrt 3$, and the three discs cover three sectors of angle $\\pi/3$, of total area $\\pi/2$.\nSo $\\sigma_2 = \\pi/\\sqrt{12}$, and the bound is sharp by [[Thue's theorem|3P246RHXMF]].\nFor $n = 3$ one gets $\\sigma_3 \\approx 0.7797$, a little above the true optimum $\\pi/\\sqrt{18} \\approx 0.7405$ of the [[Kepler conjecture|AFMBQT06ZV]].\n\n## Explanations\n\n### L1 human\n\nIn the plane, the best packing of coins puts every three neighbouring coins at the corners of an equilateral triangle.\nRogers' bound says that in any dimension, a packing of balls cannot be denser than the share of a regular simplex filled by balls at its corners.\nIn dimension $3$ this simplex is a regular tetrahedron, and the bound is close to, but above, the true answer.\n\n### L2 human\n\nRogers' bound was the best general upper bound on sphere packing density for about twenty years.\nFor large $n$ it is roughly $\\tfrac{n}{e} 2^{-n/2}$, and the Kabatiansky–Levenshtein bound improved it exponentially in 1978 by a method based on linear programming.\nIn dimensions $4$ to $36$ the Cohn–Elkies linear programming bound is better, and it is sharp in dimensions $8$ and $24$.\n",
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