Concentration of measure

Remarks

Example: take the unit sphere $S^{k}$ of $\R^{k+1}$ with the geodesic metric and the normalized surface measure. Lévy's isoperimetric inequality gives such a bound with absolute constants $C$ and $c$, so the spheres form a normal Lévy family. The Gaussian measure $\gamma_n$ on $\R^n$ concentrates with a bound that does not depend on the dimension: if $\gamma_n(A) \ge 1/2$, then $\gamma_n(A_r) \ge 1 - \frac{1}{2} e^{-r^2/2}$. Non-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 $r \le 1$. An 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 $M_f$.