Four Talagrand inequalities under the same umbrella

Understand

This note reviews the studies of the last decades emphasizing a common principle based on entropy, logarithmic Sobolev inequality and hypercontractivity, behind four most celebrated inequalities by M.

• Talagrand: the convex distance inequality, the $\mathrm {L}^1$-$\mathrm {L}^2$ variance inequality, the quadratic transportation cost inequality, and the inequality on the supremum of empirical processes.