Return to Answer

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

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.

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 AoP paper:

it is not known if the Gaussian isoperimetric inequality itself can be retrieved from optimal transport

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

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.

A corollary of this result is

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 AoP paper:

it is not known if the Gaussian isoperimetric inequality itself can be retrieved from optimal transport