We improve the constant $frac{pi}{2}$ in $L^1$-Poincare 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: $C_1le frac{pi}{2}$. 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.
In 2006 Carbery raised a question about an improvement on the naive norm inequality $|f+g|_p^p leq 2^{p-1}(|f|_p^p + |g|_p^p)$ for two functions in $L^p$ of any measure space. When $f=g$ this is an equality, but when the supports of $f$ and $g$ are d
isjoint the factor $2^{p-1}$ is not needed. Carberys question concerns a proposed interpolation between the two situations for $p>2$. The interpolation parameter measuring the overlap is $|fg|_{p/2}$. We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all $p$.
