We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume.More precisely, we show that for all even n ∈ N there exists an explicit bijection such that for every x ≠ y {0,1}n it holds that where dist(.,.) denotes the Hamming distance.In particular, this implies that the Hamming ball is bi-Lipschitz transitive. This result gives a strong negative answer to an open problem of Lovett and Viola [CC 2012], who raised the question in the context of sampling distributions in low-level complexity classes.The conceptual implication is that the problem of proving lower bounds in the context ofsampling distributions requires ideas beyond thesensitivity-based structural results of Boppana [IPL 97]. We study the mapping ψ further and show that it (and its inverse) are computable in DLOGTIME-uniform TC0, but not in AC0. Moreover, we prove that ψ is "approximately local" in the sense that all but the last output bit of ψ are essentially determined by a single input bit.