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  "body": "\n## Summary\n\nA CAT($k$) space is a [[metric space|0VKD1TP7F3]] whose geodesic triangles are at least as thin as triangles in a model surface.\nThe model surface has constant curvature $k$.\nIt is the metric version of the condition \"sectional curvature at most $k$\".\nThe notion applies to spaces with no smooth structure, such as trees, polyhedral complexes and buildings.\nCAT(0) spaces are central in geometric group theory.\n\n## Explanations\n\n### L1 human\n\nDraw a triangle in the space whose sides are shortest paths.\nDraw a triangle with the same side lengths on a flat sheet, on a sphere or on a saddle, according to $k$.\nThe space is CAT($k$) when the first triangle is never fatter than the model one: points on its sides are no farther apart than the matching points of the model.\nA tree is CAT(0): every triangle in a tree is a tripod, as thin as possible.\nA round sphere is not CAT(0), because its triangles bulge outward.\n",
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