Let $x\in S^{d-1}$ be chosen uniformly at random from the $d$-dimensional unit sphere.

I want to show that there exists a universal constant $c\in\mathbb R$ such that $\mathbb E\left[\frac{d}{||x||_1^2}\right]\le c$ for all dimensions $d$.

Any thoughts?

Let $x\in S^{d-1}$ be chosen uniformly at random from the $d$-dimensional unit sphere.

I want to show that there exists a universal constant $c\in\mathbb R$ such that $\mathbb E\left[\frac{d}{||x||_1^2}\right]\le c$ for all dimensions $d$.

Any thoughts?

Using numerical simulations, it seems that $\mathbb E\left[\frac{d}{||x||_1^2}\right]< \pi/2$, but I'm not sure how to prove it.