Area of $n$-sphere contained outside $\ell_1$ ball

Question

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 $c>0$)

Answer 1

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.