Improving constant in end-point Poincar\'e inequality on Hamming cube

Understand

We improve the constant $\frac{\pi}{2}$ in $L^1$-Poincar\'e inequality on Hamming cube.

• For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $\sqrt{\frac{\pi}{2}}$.
• For Hamming cube the sharp constant is not known, and $\sqrt{\frac{\pi}{2}}$ gives an estimate from below for this sharp constant.
• On the other hand, L.