In his book "Topics in optimal transportation", Graduate Studies in Mathematics 58, AMS 2003, Villani presents a proof, due to Gromov, of the classical isoperimetric inequality in Euclidean space using mass transportation methods. Is there a (similar) mass transportation proof towards the isoperimetric inequality for Gaussian measures?
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$\begingroup$ It may be useful to look at the paper of Cordero-Erausquin (link.springer.com/article/10.1007/s002050100185) and at the references contained therein, though I do not believe that this paper proves the Gaussian isoperimetric inequality itself (discussed a bit in Section 3). $\endgroup$– πr8Commented Jan 16, 2023 at 16:36
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$\begingroup$ I can't remember the details now, but I have it in my head that there is a transport proof of the Brunn-Minkowski inequality, and (much more hazily) that one can derive the Gaussian isoperimetric inequality from BM (or maybe Prekopa-Leindler), and so conceivably one can stitch these two observations together. I'll post again if I remember any of the relevant details. $\endgroup$– πr8Commented Jan 16, 2023 at 16:40
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See e.g. Section 2.1 "Talagrand's transport inequalities and Gaussian dimension-free concentration" of Gozlan's survey.
Theorem 2.3 there is Talagrand's result that the standard Gaussian measure on $\mathbb R^d$ satisfies Talagrand’s transport inequality $\mathbf T_2(2)$. On the other hand, Theorem 2.4 in that survey, with a very short proof, states that, for any real $C>0$, the $\mathbf T_2(C)$ property of a probability measure implies a concentration property for that measure.
It is also well known that a concentration property can be derived from an isoperimetric inequality.
On the other hand, the following is stated on p. 669 of this 2017 AoP paper (with a reference to Villani, C., 2009, Optimal Transport. Old and New):
it is not known if the Gaussian isoperimetric inequality itself can be retrieved from optimal transport
So, if there is a derivation of the Gaussian isoperimetric inequality from optimal transport, it is likely a rather recent one.
answered Jan 16, 2023 at 14:09
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$\begingroup$ Thank you, but concentration properties are only consequences of the isoperimetric statement, that is usually a more delicate issue with in particular identification of the extremal sets. $\endgroup$ Commented Jan 16, 2023 at 15:44
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$\begingroup$ @XiazhongZhu : I have added a remark on this, with further references. $\endgroup$ Commented Jan 16, 2023 at 16:29