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on $L^1$ Poincaré constant on the Hamming cube ($C_{11a}$)

The statement

$C_{11a}$ is the smallest constant such that, for every $n\ge 1$ and every function $f$ :{-1,1}^n $\to \mathbb{R}$

\mathbb{E}\bigl|f(x)-\mathbb{E}f(x)\bigr|\ \le\ C_{11a}\mathbb{E}|\nabla f|(x),

where $x=(x_{1},\dots,x_{n})$ is uniform on {-1,1}^n and

|\nabla f|(x)=\Bigl(\sum_{j=1}^n |D_{j} f(x)|^2\Bigr)^{1/2},\qquad D_{j} f(x)=\frac{f(x)-f(x^{(j)})}{2},

with $x^{(j)}=(x_{1},...,x_{j-1},-x_{j},x_{j+1},...,x_{n}).$

This is sometimes described as the (dimension-free) Cheeger constant appearing in the $L^1$ Poincaré inequality on the discrete cube.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.