The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the ndimensional Euclidean space, halfspaces have the minimal Gaussian boundary measure. Suppose we put an additional restriction on the set, that it should be symmetric about the origin. Then can we conclude that quarterspaces (intuitively the first and third quadrant in 2dimensions, say) have the minimal Gaussian boundary measure?

My guess is that the optimizer is actually a "strip"; i.e., a set of the form {$x : t \leq x_1 \leq t$}. But I'm somewhat sure that the solution to this problem is not known. You might take a look at the discussion surrounding after Corollary 3.6 in this paper by Klartag and Regev: http://eccc.hpiweb.de/report/2010/140/ Barthe may also have some relevant papers. 


Answering more @Bratt's comment than the original question: Talagrand's book people.math.jussieu.fr/~talagran/book.ps.gz Seems quite nice. 

