See e.g. Section 2.1 "Talagrand's transport inequalities and Gaussian dimension-free concentration" of [Gozlan's survey][1]. 

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. 

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On the other hand, the following is stated on p. 669 of [this 2017 AoP paper][2] (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. 

  [1]: https://www.esaim-proc.org/articles/proc/pdf/2015/04/proc145101.pdf
  [2]: https://projecteuclid.org/journals/annals-of-probability/volume-45/issue-2/Sharp-dimension-free-quantitative-estimates-for-the-Gaussian-isoperimetric-inequality/10.1214/15-AOP1072.pdf