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  "body": "\n## Description\n\nLet $n \\ge 1$ be an integer.\nFor two real [[normed spaces|03J5RAKB30]] $X$ and $Y$ of dimension $n$, the Banach–Mazur distance is\n$$d(X, Y) = \\inf \\set{\\norm{T} \\, \\norm{T^{-1}} : T \\colon X \\to Y \\text{ a linear isomorphism}}.$$\nHere $\\norm{T}$ is the [[operator norm|06EKW1AKXF]].\nThe Banach–Mazur compactum $\\mathcal{Q}_n$ is the set of isometry classes of $n$-dimensional real normed spaces with the metric $\\delta(X, Y) = \\log d(X, Y)$.\nIt is a compact [[metric space|0VKD1TP7F3]].\n\n## Remarks\n\nThe function $\\delta$ is a metric on isometry classes.\nIt is symmetric because $\\norm{T^{-1}} \\, \\norm{T}$ is the same number for $T$ and $T^{-1}$.\nThe triangle inequality comes from $\\norm{ST} \\le \\norm{S} \\, \\norm{T}$, so $d(X, Z) \\le d(X, Y) \\, d(Y, Z)$.\nIf $d(X, Y) = 1$, then a compactness argument gives an isomorphism with $\\norm{T} \\, \\norm{T^{-1}} = 1$, and a multiple of it is an isometry.\n\nBy John's theorem, $d(X, \\ell_2^n) \\le \\sqrt{n}$ for every $n$-dimensional normed space $X$, where $\\ell_2^n$ is $\\R^n$ with the Euclidean norm.\nSo the diameter of $\\mathcal{Q}_n$ in the multiplicative form is at most $n$.\nGluskin proved that this order is right: there are $n$-dimensional spaces at distance at least $c n$ from each other, for a constant $c \u003e 0$ that does not depend on $n$.\n\n## Explanations\n\n### L1 human\n\nEach point of the Banach–Mazur compactum is a way to measure length in $n$-dimensional space, up to a change of coordinates that keeps lengths.\nTwo such ways are close when a linear change of coordinates can squeeze one unit ball between the other unit ball and a slightly larger copy of it.\nFor $n = 2$, the square ball of the max norm and the round disc of the Euclidean norm are at distance $\\log \\sqrt{2}$.\n",
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