Beyond Enflo’s problem

Enflo's problem deals with Pisier–Poincaré type inequality for functions of Rademacher random variables. The problem was solved recently by P. Ivanisvili, R. Van Handel and me. However, this type of Poincaré inequality is only one out of the whole scale of inequalities filling the gap between Pisier's inequality and singular integral inequalities on Hamming cube. It is important and interesting: a) to understand what the sharp constants in these results are; b) what is the stock of Banach spaces X such that the results hold for functions of Rademacher random variables with values in X. In the talk I will explain this and the relations with the Ribe program and Bourgain's discretization theorem.