Talagrand's convex distance inequality
= Talagrand's convex distance inequality
{c}
{title2=$\mathbb P(A)\mathbb P(d_T(X,A)\geq s)\leq e^{-s^2/4}$}
On a finite product of standard <probability spaces>, for any measurable event $A$ of positive probability,
$$
\mathbb E\exp(d_T(X,A)^2/4)\leq1/\mathbb P(A).
$$
Consequently $\mathbb P(A)\mathbb P(d_T(X,A)\geq s)\leq e^{-s^2/4}$. Here $d_T$ is the <Talagrand convex distance>. The distance-event conclusion can be read with outer probability when necessary.