How do classical concentration inequalities extend to functions taking values in normed spaces? Such questions arise in various settings (in functional analysis, random matrix theory, etc.), and may be studied on a case-by-case basis. On the other hand, in the Gaussian case, Pisier discovered a remarkable principle that explains in complete generality how concentration inequalities for vector-valued nonlinear functions is controlled by the corresponding behavior of linear functions.
Unfortunately, Pisier's argument appears to be very specific to the Gaussian distribution. In particular, its obvious analogue on the discrete cube, which would be of interest in many applications, is known to be false. In recent work with Paata Ivanisvili and Sasha Volberg, we discovered a less obvious formulation of Pisier's inequality that provides a sharp analogue of the linear to nonlinear principle on the discrete cube. This enabled us to settle an old problem in the geometry of Banach spaces due to Enflo (1978). I will aim to explain this new inequality, where it comes from, some applications, and some remaining mysteries.
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