  Let $(X,d)$ be a compact metric space and $\mu$ a Borel probability on $X$.
For each $N\geq 1$ let $d^N_\infty$ be the $\ell_\infty$-product on $X^N$ of
copies of $d$, and consider $1$-Lipschitz functions $X^N\to\mathbb{R}$ for
$d^N_\infty$.
  If the support of $\mu$ 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.
