Gaussian Isoperimetric Inequality

You have probably heard of the classical isoperimetric problem: among all closed curves in the plane with equal perimeter, which curve encloses the largest area? Equivalently, the question can be formulated as the following: among all curves in the plane enclosing the same area, which one, if any, has the smallest perimeter. The answer to this question is the circle, and this is true even in a much more general setting. Among all Borel sets in Rn with equal Lebesgue measure, balls have the smallest “boundary measure”. One can ask similar question about the Gaussian measure in Rn, which leads to the Gaussian isoperimetric inequality. This time, the solution is not balls but halfspaces. Before formally stating and proving the Gaussian isoperimetric inequality, let me first define a few terms that will be useful in the rest of this write-up. First, for x 2 Rn, the probability density function of the standard Gaussian distribution jn(x) is: jn(x) = 1 p 2p exp( kxk2/2)