Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?

Question

Let $X$ be a real-valued standard normal variable. Then, for any differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $E[f(X)^2] < \infty$ and $E[\bigl( f'(X) \bigr)^2] < \infty$, it is well-known that \begin{equation} \text{Var}(f(X)) \leq E[\bigl(f'(X)\bigr)^2] \end{equation} and is called the Gaussian Poincare inequality.

I can see that this is the Poincare inequality with $p=2$ according to the Wikipedia article https://en.wikipedia.org/wiki/Poincar%C3%A9_inequality

Now, I wonder if the Gaussian Poincare inequality holds for $p=1$. That is, do we also have \begin{equation} E[\bigl \lvert f(X) - E[f(X)] \bigr \rvert ] \leq E[\bigl \lvert f'(X) \bigr \rvert ] \end{equation} in general?

In the link above, the generalization of the Poincare inequality to general measure spaces is considered as well. I searched for papers myself but was not able to find anything specialized to Gaussian measures.

Could anyone please help me?

Answer 1

Yes, Gaussians also satisfy a Poincaré inequality with $p = 1$ (such an inequality is equivalent to what is called a "Cheeger inequality"). More generally, E. Milman has shown that for log-concave measures, all $(p, q)$-Poincaré inequalities are equivalent:

Milman, E. On the role of convexity in isoperimetry, spectral gap and concentration. Invent. math. 177, 1–43 (2009). https://doi.org/10.1007/s00222-009-0175-9