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  "body": "\n## Summary\n\nA nonexpansive map between metric spaces is a map that never increases distances: $d(f(x), f(y)) \\le d(x, y)$.\nThese are exactly the maps that are [[Lipschitz|0AW1E6YTJY]] with constant $1$; they are also called short maps or metric maps.\nThey include isometries and [[contraction mappings|0QN69YA357]].\nIn fixed point theory they are the borderline case where the Banach fixed point theorem no longer applies and the geometry of the space matters.\n\n## Explanations\n\n### L1 human\n\nA nonexpansive map may move points around, but it never pulls two points further apart.\nRotations of the plane, reflections, and the map sending each point to the nearest point of a closed disc are all nonexpansive.\nThe map $x \\mapsto 2x$ on the real line is not, since it doubles distances.\nUnlike contractions, nonexpansive maps need not have fixed points: the shift $x \\mapsto x + 1$ on the line moves every point.\n",
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