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  "body": "\n## Summary\n\nThe Fréchet mean generalises the average to data in a [[metric space|0VKD1TP7F3]]: it is a point that minimises the expected squared distance to the data.\nIn Euclidean space it is the usual mean.\nIt also makes sense for points on a sphere, on a [[Riemannian manifold|7CBT4SM83Y]], or in spaces of trees, shapes or covariance matrices.\nIn these spaces adding and dividing points is not defined.\nIt need not exist or be unique, and the conditions for uniqueness involve the curvature of the space.\nIt is named after Maurice Fréchet; local minimisers on Riemannian manifolds are also called Karcher means.\n\n## Explanations\n\n### L1 human\n\nTo average two cities on the globe, you cannot average their coordinates in 3D space, since the result lies inside the Earth.\nInstead, look for the place on the surface that minimises the sum of squared distances, measured along the surface, to the two cities.\nThat place is the midpoint of the shortest route between them.\nThe same recipe works for averaging directions, rotations or shapes.\nFor two exactly opposite points on a sphere, every point of the great circle halfway between them is a Fréchet mean, so the mean is not unique.\n",
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