We improve the constant π/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 π/2 (Maurey– Pisier). For Hamming cube the sharp constant is not known, and the square root of π/2 gives an estimate from below for this sharp constant. On the other hand, Ben Efraim and Lust-Piquard have shown an estimate from above: π/2 without the square root. There are at least two other independent proofs of the same estimate from above. 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 their constant is sharp. But here we give a better estimate from above, showing that Poincar´e constant C1 is strictly smaller than Ben Efraim and Lust-Piquard constant π/2. It is still not clear whether C1 > pπ/2. The proof of Ben Efraim and Lust-Piquard used non-commutative harmonic analysis to prove their estimate. Our approach is different. We discuss this circle of questions, their relation with isoperimetric inequalities on Hamming cube and with Margoulis sharp threshold network theorem.
