Let me recall subj:

If $s>0$, $A$ and $B$ are two subsets of $\mathbb{S}^{n}$, $|A|=|B|$ ($|\cdot|$ stands for the Lebesgue measure on the sphere) and $B$ is a cup $B=\{ (x_1,x_2,\dots,x_n)\in \mathbb{S}^n, x_n\leq t \}$ (for some $t\in [-1,1]$), then $|A_s|\geq |B_s|$, where $A_s$ means $s$-neighborhood of the set.

It leads to measure concentration inequalities for the sphere and so has numerous applications. So I guess that Levy's initial proof was simplified, maybe not once. What is the easiest proof of the inequality and where to read it?