Optimal constants in concentration inequalities on the sphere and in the Gauss space

TL;DR

This paper establishes concentration inequalities on the sphere and Gaussian space with optimal constants, providing sharp deviation bounds for Lipschitz functions.

Contribution

It derives new, sharper concentration bounds with optimal constants for Lipschitz functions on the sphere and Gaussian space, improving existing results.

Findings

Derived one-sided and two-sided deviation bounds for Lipschitz functions.

Established bounds with optimal constants that are more elegant and slightly better than previous results.

Provided explicit bounds for functions on the sphere and Gaussian space with median and mean.

Abstract

We show several variants of concentration inequalities on the sphere stated as subgaussian estimates with optimal constants. For a Lipschitz function, we give one-sided and two-sided bounds for deviation from the median as well as from the mean. For example, we show that if $\mu$ is the normalized surface measure on $S^{n-1}$ with $n\geq 3$, $f : S^{n-1} \to \mathbb{R}$ is $1$-Lipschitz, $M$ is the median of $f$, and $t >0$, then $\mu\big(f \geq M +t\big) \leq \frac 12 e^{-nt^2/2}$. If $M$ is the mean of $f$, we have a two-sided bound $\mu\big(|f - M| \geq t\big) \leq e^{-nt^2/2}$. Consequently, if $\gamma$ is the standard Gaussian measure on $\mathbb{R}^n$ and $f : \mathbb{R}^{n} \to \mathbb{R}$ (again, $1$-Lipschitz, with the mean equal to $M$), then $\gamma \big(|f - M| \geq t\big) \leq e^{-t^2/2}$. These bounds are slightly better and arguably more elegant than those available elsewhere in the literature.