I'm reading a few chapters of An elementary introduction to modern convex geometry, Flavors of geometry by Keith Ball for a class and at one point there's a theorem
Let $A\subset \mathbb{R}^n$ be a measurable set and $\mu$ be the standard Gaussian measure on $\mathbb{R}^n$. Then $\int e^{{}^{d(x,A)^2}/_4} \mathrm{d}\mu \leq \frac{1}{\mu(A)}$. Consequently, if $\mu(A)= \frac{1}{2}$ then $\mu(A_\epsilon)\geq 1 - 2e^{-^{\epsilon^2}/_4}$.
I understand the proof and the statement of the theorem, but I don't understand the interpretation that's given later, "There is an Euclidean ball of some radius $R$ whose Gaussian measure is $\frac{1}{2}$. According to the theorem above, Gaussian measure concentrates near the boundary of this ball".
Could you explain why this theorem shows that the measure concentrates "near the boundary", and not simply "just around the ball" ?
Because to me it seems that, calling $B$ the mentioned ball, most of the measure is in $B$, and then almost all the measure is in $B_\epsilon$, but there's definitely less measure "near the boundary" (which I interpret as "in $B_\epsilon \setminus B$) than in $B$.
Note $A_\epsilon$ is the set of points at distance less than $\epsilon$ of $A$, aka $A+\epsilon B_2^n$ where $B_2^n$ is the standard euclidean ball in dimension $n$