Talagrand's influence inequality (1994) is an asymptotic improvement of the classical Poincaré inequality on the Hamming cube with numerous applications to Boolean analysis, discrete probability theory and geometric functional analysis. In this talk, we shall introduce a metric space-valued version of Talagrand's inequality and show its validity for some natural classes of spaces. Emphasis will be given to the probabilistic aspects of the proofs. We will also explain a geometric application of this metric invariant to the bi-Lipschitz embeddability of a natural family of finite metrics and mention related open problems. The talk is based on joint work with D. Cordero-Erausquin.
