  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. Ben Efraim and F. Lust-Piquard have shown
an estimate from above: $C_1\le \frac{\pi}{2}$. There are at least two other
independent proofs of the same estimate from above (we write down them in this
note). Since those proofs are very different from the proof of Ben Efraim and
Lust-Piquard but gave the same constant, that might have indicated that
constant is sharp. But here we give a better estimate from above, showing that
$C_1$ is strictly smaller than $\frac{\pi}{2}$. It is still not clear whether
$C_1> \sqrt{\frac{\pi}{2}}$. We discuss this circle of questions and the
computer experiments.
