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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and $\\mu$ a Borel [[probability measure|059TJN0NZ1]] on $X$.\nFor a Borel set $A \\subseteq X$ and $r \u003e 0$, let $A_r = \\set{x \\in X : d(x, A) \u003c r}$.\nThe *concentration function* of $(X, d, \\mu)$ is\n$$\\alpha(r) = \\sup \\set{1 - \\mu(A_r) : A \\subseteq X \\text{ Borel}, \\ \\mu(A) \\ge 1/2}, \\qquad r \u003e 0.$$\nA sequence $(X_k, d_k, \\mu_k)$ of such spaces exhibits *concentration of measure*, and is called a *normal Lévy family*, if there are constants $C, c \u003e 0$ with\n$$\\alpha_k(r) \\le C e^{-c k r^2} \\quad \\text{for all } k \\ge 1 \\text{ and } r \u003e 0.$$\n\n## Remarks\n\nExample: take the unit sphere $S^{k}$ of $\\R^{k+1}$ with the geodesic metric and the normalized surface measure.\nLévy's isoperimetric inequality gives such a bound with absolute constants $C$ and $c$, so the spheres form a normal Lévy family.\nThe [[Gaussian measure|P0DV2TCH1N]] $\\gamma_n$ on $\\R^n$ concentrates with a bound that does not depend on the dimension: if $\\gamma_n(A) \\ge 1/2$, then $\\gamma_n(A_r) \\ge 1 - \\frac{1}{2} e^{-r^2/2}$.\nNon-example: the two-point space $\\set{0, 1}$ with $d(0, 1) = 1$ and $\\mu(\\set{0}) = \\mu(\\set{1}) = 1/2$; for $A = \\set{0}$ and $r \\le 1$ one has $A_r = A$, so $\\alpha(r) \\ge 1/2$ for $r \\le 1$.\nAn equivalent form: $\\mu(\\set{x : \\abs{f(x) - M_f} \\ge r}) \\le 2 \\alpha(r)$ for every $1$-Lipschitz function $f \\colon X \\to \\R$ with median $M_f$.\n",
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