I need to compute probabilities of the form $P( \Vert X \Vert_\infty < r ),$ where $X$ is a random variable of dimension $n$, drawn with a uniform distribution on the unit sphere $\mathcal{S}_{n-1}$. Clearly, the distribution of $Z = \Vert X \Vert_\infty$ is supported by the interval $[\frac{1}{\sqrt{n}},1]$.

Any hint to get a simple formula (ideally a closed-form expression or a one-dimensional integral) will be greatly appreciated !

**Note:**
Using the fact that $X \sim \frac{Y}{\Vert Y\Vert_2}$ for $Y \sim \mathcal{N}(0,1)$, at first I thought I could first compute the probabilities
$P_i:=P( |X_i| < r \Vert X \Vert_2)$, which can be expressed as a quantile of the F-distribution, but then I realized that those events **are not independent**, so
$P( \Vert X \Vert_\infty < r ) \neq \prod_{i=1}^n P_i$.