For a given $r>1$, what is the surface area of $\mathbb S^{n-1}$ (the sphere of radius 1 in $\mathbb R^n$) which is contained outside of the $\ell\_1$ ball of radius $r$? Or equivalently, if $X\sim U(\mathbb S^{n-1})$, a point sampled uniformly from the sphere, what is the probability that $\Vert X\Vert\_1\geq r$?


This is easy to compute for $r\geq \sqrt{n-1}$, as the area is exactly $2^n$ spherical caps, and this has a clean, closed-form formula. For smaller values of $r$, however, these caps intersect, and the algebra gets worse.


The exact value of this probability matters less than approximate asymptotic bounds for $n$ large, with $r$ growing in $n$ (like $n^c$ for $c>0$)



Area of $n$-sphere contained outside $\ell_1$ ball
We use the gaussian construction of $U(\mathbb{S}^{n-1})$ to write $$\|X\|\_{\ell^1} = \frac{\sum\_i |N\_i|}{\sqrt{\sum\_i |N\_i|^2}}$$ with $N\_i$ iid random gaussian variables with variance 1. We have $$\mathbb{P}(\|X\|\_{\ell^1} \leq \alpha\sqrt{n})\leq \mathbb{P}(\sum\_i |N\_i| \leq \alpha\_1 n) + \mathbb{P}(\sum\_i |N\_i|^2 \geq \alpha\_2 n)$$ for all $\alpha\_1,\alpha\_2 $ such that $\frac{\alpha\_1}{\sqrt{\alpha\_2}}\geq \alpha$. Then if $\alpha<\mathbb{E}(|N\_1|)=\sqrt{\frac{2}{\pi}}$ you can choose $\alpha\_2 >1$ and $\alpha\_1 <\mathbb{E}(|N\_1|)$ to write a large deviation principle (Cramer theorem). The upper bound is
$$\mathbb{P}(\sum\_i |N\_i| \leq \alpha\_1 n)\leq \Big(\frac{\mathbb{E}(e^{\lambda|N\_1|})}{e^{\alpha\_1 \lambda}}\Big)^n $$for any $\lambda$. Because $\alpha\_1<\mathbb{E}(|N|\_1)$ you can find $\lambda$ such that $\frac{\mathbb{E}(e^{\lambda|N\_1|})}{e^{\alpha\_1 \lambda}}=\kappa\_{\alpha\_1}<1$. We do the same for $|N\_i|^2$ and get $$\mathbb{P}(\sum\_i |N\_i|^2 \geq \alpha\_2 n)\leq (\kappa\_{\alpha\_2})^n $$ with $\kappa\_{\alpha\_2}<1$. 


Conclusion for $\alpha<\mathbb{E}(|N\_1|)$ we have that $$\mathbb{P}(\|X\|\_{\ell^1} \leq \alpha\sqrt{n})\leq (\kappa\_{\alpha\_1})^n+(\kappa\_{\alpha\_2})^n$$ decays exponentially. 


