Motivated by random walks on subsets of the hypercube, we prove two discrete functional inequalities on the hypercube. First, we give a short, elementary proof of the Poincar\'e inequality on increasing subsets of the cube recently established by Fei and Ferreira Pinto Jr, which yields an $O(n^2)$ upper bound on the mixing time of censored random walks, improving upon previous bounds. Second, adapting Samorodnitsky's induction method to the $p$-biased setting, we establish a sharp $p$-biased edge-isoperimetric inequality for real-valued increasing functions, which recovers the classic biased edge-isoperimetric inequality for increasing sets and identifies increasing subcubes as the extremizers. This result also admits a probabilistic interpretation in terms of maximizing the mean first exit time of biased random walks.
Categories: math.CO math.PR