The Gaussian correlation inequality (GCI), proven by Royen in 2014, states that any two centrally symmetric convex sets (say K and L) in the Gaussian space are positively correlated. We will prove a new quantitative version of the GCI which gives a lower bound on this correlation based on the "common influential directions" of K and L. This can be seen as a Gaussian space analogue of Talagrand's well known correlation inequality for monotone functions. To obtain this inequality, we propose a new approach, based on analysis of Littlewood type polynomials, which gives a recipe to transfer qualitative correlation inequalities into quantitative correlation inequalities. En route, we also give a new notion of influences for convex symmetric sets over the Gaussian space which has many of the properties of influences from Boolean functions over the discrete cube. Much remains to be explored, in particular, about this new notion of influences for convex sets.
Based on joint work with Shivam Nadimpalli and Rocco Servedio.