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  "body": "\n## Summary\n\nThe Banach–Mazur distance measures how far two [[normed vector spaces|03J5RAKB30]] of the same finite dimension are from being isometric.\nIt is the smallest possible distortion $\\norm{T} \\norm{T^{-1}}$ of a [[linear isomorphism|0B8T10CE7N]] $T$ between them.\nIn geometric terms, it compares the unit balls up to linear maps.\nWith it, the isometry classes of $n$-dimensional normed spaces form the [[Banach–Mazur compactum|RH1WTN38RQ]].\nMany questions of local Banach space theory and asymptotic convex geometry are estimates of this distance.\n\n## Explanations\n\n### L1 human\n\nA norm on $\\R^n$ is described by its unit ball, a symmetric convex body.\nTwo norms are close in the Banach–Mazur sense if a linear change of coordinates makes the unit balls nearly equal.\nThen one unit ball fits inside the other, and the other fits inside a slightly enlarged copy of the first.\nThe distance is $1$ exactly when the two spaces are isometric.\nFor instance, the square ball of the maximum norm on $\\R^2$ and the round disc of the Euclidean norm are at distance $\\sqrt{2}$.\nThe disc fits in the square, and the square fits in the disc enlarged by $\\sqrt{2}$.\n",
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