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dohmatob
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Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition $$ \operatorname{Hess}_x(V) \succeq cI_d,\text{ for all }x \in \mathbb R^d. $$

Now, It is known that a distribution $\mu$ on $\mathbb R^d$ which has finite moment and density not supported on an affine subspace can be approximated with a log-concave distribution $\nu$ (Lemma 2.1 of this paper).

Definition: $\mu$ is said to be log-convave with constant $c > 0$, if density $d\mu \propto e^{-V}dvol$ satisfying the curvature condition $$ \operatorname{Hess}_x(V) \succeq cI_d,\text{ for all }x \in \mathbb R^d. $$

Question

Would such a distribution $\mu$ then satisfy a transportation-cost inequality for the Wasserstein $2$-distance (see here for definitions definitions, just in case) ?

It is known that a distribution $\mu$ on $\mathbb R^d$ which has finite moment and density not supported on an affine subspace can be approximated with a log-concave distribution (Lemma 2.1 of this paper).

Definition: $\mu$ is said to be log-convave with constant $c > 0$, if density $d\mu \propto e^{-V}dvol$ satisfying the curvature condition $$ \operatorname{Hess}_x(V) \succeq cI_d,\text{ for all }x \in \mathbb R^d. $$

Question

Would such a distribution $\mu$ then satisfy a transportation-cost inequality for the Wasserstein $2$-distance (see here for definitions, just in case) ?

Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition $$ \operatorname{Hess}_x(V) \succeq cI_d,\text{ for all }x \in \mathbb R^d. $$

Now, It is known that a distribution $\mu$ on $\mathbb R^d$ which has finite moment and density not supported on an affine subspace can be approximated with a log-concave distribution $\nu$ (Lemma 2.1 of this paper).

Question

Would such a distribution $\mu$ then satisfy a transportation-cost inequality for the Wasserstein $2$-distance (see here for definitions, just in case) ?

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dohmatob
dohmatob
  • 6.9k
  • 1
  • 18
  • 76

It is known that a distribution $\mu$ on $\mathbb R^d$ which has finite moment and density not supported on an affine subspace can be approximated with a log-concave distribution (Lemma 2.1 of this paper).

Definition: $\mu$ is said to be log-convave with constant $c > 0$, if density $d\mu \propto e^{-V}dvol$ satisfying the curvature condition $$ \operatorname{Hess}_x(V) \succeq cI_d,\text{ for all }x \in \mathbb R^d. $$

Question

Would such a distribution $\mu$ then satisfy a transportation-cost inequality for the Wasserstein $2$-distance (see here for definitions, just in case) ?

It is known that a distribution $\mu$ on $\mathbb R^d$ which has finite moment and density not supported on an affine subspace can be approximated with a log-concave distribution (Lemma 2.1 of this paper).

Question

Would such a distribution $\mu$ then satisfy a transportation-cost inequality for the Wasserstein $2$-distance (see here for definitions, just in case) ?

It is known that a distribution $\mu$ on $\mathbb R^d$ which has finite moment and density not supported on an affine subspace can be approximated with a log-concave distribution (Lemma 2.1 of this paper).

Definition: $\mu$ is said to be log-convave with constant $c > 0$, if density $d\mu \propto e^{-V}dvol$ satisfying the curvature condition $$ \operatorname{Hess}_x(V) \succeq cI_d,\text{ for all }x \in \mathbb R^d. $$

Question

Would such a distribution $\mu$ then satisfy a transportation-cost inequality for the Wasserstein $2$-distance (see here for definitions, just in case) ?

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dohmatob
dohmatob
  • 6.9k
  • 1
  • 18
  • 76

Does log-concave approximable distribution satisfy transportation-cost inequality?

It is known that a distribution $\mu$ on $\mathbb R^d$ which has finite moment and density not supported on an affine subspace can be approximated with a log-concave distribution (Lemma 2.1 of this paper).

Question

Would such a distribution $\mu$ then satisfy a transportation-cost inequality for the Wasserstein $2$-distance (see here for definitions, just in case) ?