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  "body": "\n## Definition\n\nLet $(M, d)$ be a [[metric space|0VKD1TP7F3]] and let $X$ be a random element of $M$ with $E[d(X, q)^2] \u003c \\infty$ for some $q \\in M$.\nThe *Fréchet function* of $X$ is $F(p) = E[d(X, p)^2]$ for $p \\in M$, an [[expected value|0WQNRCNY3M]].\nA *Fréchet mean* of $X$ is a point $\\mu \\in M$ with\n$$F(\\mu) = \\inf_{p \\in M} F(p).$$\nFor points $x_1, \\dots, x_n \\in M$ and weights $w_i \u003e 0$, a *Fréchet mean* is a minimiser of $p \\mapsto \\sum_{i=1}^{n} w_i \\, d(x_i, p)^2$; with equal weights it is the *sample Fréchet mean*.\n\n## Remarks\n\nThe condition on $q$ and the triangle inequality give $F(p) \u003c \\infty$ for every $p$.\nThe set of Fréchet means may be empty or have several points.\nOn a complete Riemannian manifold a local minimiser of $F$ is called a Karcher mean.\nReplacing $d^2$ by $d$ gives the Fréchet median.\n\nExample: in $M = \\R^k$ with the Euclidean distance, $\\sum_i \\norm{x_i - p}^2 = \\sum_i \\norm{x_i - \\bar x}^2 + n \\norm{\\bar x - p}^2$, where $\\bar x$ is the [[sample mean|03Y6AQ2S0J]].\nSo the sample Fréchet mean is unique and equals $\\bar x$.\n\nNon-uniqueness: on the unit sphere with the great-circle distance, take two antipodal points $x$ and $-x$.\nIf $p$ is at distance $\\theta$ from $x$, it is at distance $\\pi - \\theta$ from $-x$, and $\\theta^2 + (\\pi - \\theta)^2$ is smallest exactly at $\\theta = \\pi/2$.\nSo every point of the equator between them is a Fréchet mean, and the \"midpoint\" is not a single point.\n\nNon-example: if $X$ is real with the Cauchy distribution, then $E[(X - q)^2] = \\infty$ for every $q \\in \\R$.\nThe moment condition fails, $F$ is infinite everywhere, and $X$ has no Fréchet mean, just as it has no ordinary mean.\n",
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