Concentration inequalities, the entropy method, search for "super"-concentration

The entropy method has proved to be a handy device to derive concentration inequalities for functions of independent random variables. So far, it has delivered the tightest general inequalities concerning suprema of bounded empirical processes, conditional Rademacher averages or self-bounded functionals like VC-entropies. It has also been pointed out that off-the-shelf exponential Efron-Stein inequalities delivered by the entropy method may not be as tight as they should, they may fail to capture the so-called super-concentration phenomenon. We show that in the simplest setting of this phenomenon, suprema of Gaussian vectors, complementing the entropy method with appropriate representations provides a pedestrian derivation of tight "super"-concentration inequalities.