2018/11/14 by Ivanisvili, Paata, Li, Dong, van Handel, Ramon +1
#26D15 #42B20 #42B35 #47A30 #60E15 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #F.2.2 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1811.05584
We improve the constant \fracπ2 in L1-Poincaré inequality on Hamming cube. For Gaussian space the sharp constant in L1 inequality is known, and it is √\fracπ2. For Hamming cube the sharp constant is not known, and √\fracπ2 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: C1≤ \fracπ2. 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 C1 is strictly smaller than \fracπ2. It is still not clear whether C1> √\fracπ2. We discuss this circle of questions and the computer experiments.