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 the square root of $\frac{\pi}{2}$ (Maurey—Pisier). For Hamming cube the sharp constant is not known, and the square root of $\frac{\pi}{2}$ gives an estimate from below for this sharp constant. On the other hand, Ben Efraim and...