Discrete logarithmic Sobolev inequalities in Banach spaces

We shall discuss certain aspects of vector-valued harmonic analysis on the discrete hypercube. After presenting the geometric motivation behind such investigations, we will survey known results on the Poincaré inequality and Talagrand’s influence inequality. Then we will proceed to present a new optimal vector-valued logarithmic Sobolev inequality in this context. The talk is based on joint work with D. Cordero-Erausquin (Sorbonne).