Concentration of volume towards the boundary Consider a Euclidean space $X$ of large dimension $N$. For a measurable subset $G\subseteq X$ and $\varepsilon>0$ let
$$G_\varepsilon:=\{x\in G\mid B_\varepsilon(x)\subseteq G\}$$
be the set of all points in $G$ with distance at least $\varepsilon$ from the boundary of $G$.
I am looking for results of the form
$$\mathrm{Vol}(G_\varepsilon)/\mathrm{Vol}(G)\leq \exp(-N\rho(\varepsilon))$$
for bounded measurable sets (or similar results).
This is to make precise the intuition that, as the dimension of the space increases a larger fraction of the volume of a set is concentrated near the boundary.
For example let $G=[0-\varepsilon,1+\varepsilon]^N$, then
$G_\varepsilon=[0,1]^N$ and the volume fraction above becomes $(1+2\varepsilon)^{-N}$ hence the conjecture for exponential decrease.
Are there any results available already?
 A: $\newcommand\ep\varepsilon\newcommand\R{\mathbb R}$Suppose that $G$ is a measurable subset of $\R^N$ with volume $|G|>0$ such that
$$|G|\le C^N|B|,$$
where $C>0$ is a real constant and $B$ is the unit ball in $\R^N$.
Without loss of generality (wlog), $|G_\ep|>0$. Also, $G_\ep+\ep B\subseteq G$ for any real $\ep>0$. So, in view of the Brunn--Minkowski inequality,
$$|G|\ge|G_\ep+\ep B|\ge(|G_\ep|^{1/N}+\ep|B|^{1/N})^N
=|G_\ep|\Big(1+\ep\Big(\frac{|B|}{|G_\ep|}\Big)^{1/N}\Big)^N
\ge|G_\ep|\Big(1+\ep\Big(\frac{|B|}{|G|}\Big)^{1/N}\Big)^N
\ge|G_\ep|\Big(1+\frac\ep C\Big)^N.
$$
Thus,
$$\frac{|G_\ep|}{|G|}\le e^{-N\rho(\ep)},\tag{1}\label{1}$$
where $\rho(\ep):=\ln(1+\frac\ep C)$.

Here is another (better?) statement:
Suppose that $|G|>0$ and the values of the (outer) measure of the projections of $G$ onto the $N$ coordinate axes of $\R^N$ are no greater than positive numbers $L_1,\dots,L_N$, respectively. Then
$$\frac{|G_\ep|}{|G|}\le\prod_{i=1}^N(1-2\ep/L_i)_+,$$
where $u_+:=\max(0,u)$.
For $N=1$, this follows almost immediately from the definition of the outer measure. For $N>1$, this follows by induction, using one-dimensional cross-sections and the Tonelli theorem.
In particular, if $L_i=L>0$ for all $i$, then
$$\frac{|G_\ep|}{|G|}\le(1-2\ep/L)_+^N\le e^{-2N\ep/L}. \tag{2}\label{2}$$
A further improvement can be obtained by replacing $L_i$, for each $i$, by an upper bound on the measures of all one-dimensional cross-sections of $G$ in the direction of the $i$th coordinate axis.
By considering $G=CB$ and $G=[0,L]^N$, it is clear that the bounds in \eqref{1} and \eqref{2} are optimal, each in its own way.
