The Gaussian correlation inequality (GCI), proved by Royen in 2014, states that any two centrally symmetric convex sets (say K and L) in Gaussian space are positively correlated. We establish 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 Boolean functions.
To obtain this inequality, we propose a new approach, based on analysis of Littlewood type polynomials, which gives a recipe for transferring qualitative correlation inequalities into quantitative correlation inequalities. En route, we also give a new notion of influences for symmetric convex symmetric sets over Gaussian space which has many of the properties of influences of Boolean functions over the discrete cube. Much remains to be explored about this new notion of influences for convex sets.
Based on joint works with Anindya De and Shivam Nadimpalli.