## Volume of $n$-sphere contained in $\ell_1$ ball

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.