On the failure of concentration for the ℓ _∞-ball

Let (X, d) be a compact metric space and µ a Borel probability on X. For each N ≥ 1 let d_∞ ^N be the ℓ_∞-product on X^N of copies of d, and consider 1-Lipschitz functions X^N → ℝ for d_∞ ^N. If the support of µ is connected and locally connected, then all such functions are close in probability to juntas: that is, functions that depend on only a few coordinates of X^N. This describes the failure of measure concentration for these product spaces, and can be seen as a Lipschitz-function counterpart of the celebrated result of Friedgut that Boolean functions with small influences are close to juntas.