SHARP DIMENSION FREE QUANTITATIVE ESTIMATES FOR THE GAUSSIAN ISOPERIMETRIC INEQUALITY

Abstract. We provide a full quantitative version of the Gaussian isoperimetric in-equality: the difference between the Gaussian perimeter of a given set and a half-space with the same mass controls the gap between the norms of the corresponding barycenters. In particular, it controls the Gaussian measure of the symmetric dif-ference between the set and the half-space oriented so to have the barycenter in the same direction of the set. Our estimate is independent of the dimension, sharp on the decay rate with respect to the gap and with optimal dependence on the mass. 2010 Mathematics Subject Class. 49Q20, 60E15. 1.