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  "body": "\n## Summary\n\nThe Carathéodory metric, or Carathéodory pseudodistance, of a complex manifold $X$ uses holomorphic maps from $X$ to the unit disc.\nThe distance of two points is the largest Poincaré distance between their images under such maps.\nIt is the smallest pseudodistance for which all holomorphic maps from $X$ to the disc are distance-decreasing, and it is a biholomorphic invariant.\nIt is always at most the [[Kobayashi metric|DY8QQ5GTTX]], and on bounded convex domains the two agree by [[Lempert's theorem|2PANNSQ3G6]].\nConstantin Carathéodory introduced it in the 1920s for domains in $\\C^n$.\n\n## Explanations\n\n### L1 human\n\nBounded holomorphic functions separate the points of a bounded domain.\nThe Carathéodory distance uses them as rulers: map the domain into the unit disc by a holomorphic function and measure the hyperbolic distance of the two image points.\nThe largest value over all such maps is the Carathéodory distance.\nOn $\\C$ every bounded holomorphic function is constant by Liouville's theorem, so the Carathéodory distance of $\\C$ is $0$.\nOn the unit disc it is the hyperbolic distance.\n",
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