On the failure of concentration for the \ell_\infty-ball

TL;DR

This paper investigates the failure of measure concentration in 0-product spaces of a compact metric space, showing that Lipschitz functions are close to juntas under certain topological conditions, paralleling Friedgut's Boolean function results.

Contribution

It establishes conditions under which measure concentration fails for 0-product spaces, linking Lipschitz functions to juntas and extending Friedgut's influence-based results.

Findings

Lipschitz functions on product spaces are close to juntas when the support is connected.

The result describes a failure of measure concentration in these spaces.

It provides a Lipschitz-function analogue of Friedgut's theorem on Boolean functions.

Abstract

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.