Sharp convergence rates for averaged nonexpansive maps

We establish sharp estimates for the convergence rate of the Kranosel’skiĭ–Mann fixed point iteration in general normed spaces, and we use them to show that the optimal constant of asymptotic regularity is exactly 1/π. To this end we consider a nested family of optimal transport problems that provide a recursive bound for the distance between the iterates. We show that these bounds are tight by building a nonexpansive map T: [0, 1]^N → [0, 1]^N that attains them with equality, settling a conjecture by Baillon and Bruck. The recursive bounds are in turn reinterpreted as absorption probabilities for an underlying Markov chain which is used to establish the tightness of the constant 1/π.