TITLE: An extension of Gaussian Isoperimetry
QUESTION [9 upvotes]: The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces 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 quarter-spaces (intuitively the first and third quadrant in 2-dimensions, say) have the minimal Gaussian boundary measure?

REPLY [3 votes]: 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.hpi-web.de/report/2010/140/
Barthe may also have some relevant papers.