Alexander Volberg: Poincaré inequalities on Hamming cube: analysis, combinatorics, probability

Abstract: We improve the constant π/2 in L^1-Poincaré inequality on Hamming cube. For Gaussian space the sharp constant in L^1 inequality is known, and it is sqrt(π/2) (Maurey-Pisier). For Hamming cube the sharp constant is not known, and sqrt(π/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 less or equal π/ 2 .
There are at least two other proofs of the same estimate from above (we write down one of 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 π/2 . It is still not clear whether C1 greater than sqrt(π/2).
We discuss this circle of questions.