Guarantee for closest random point on a sphere

Question

Let $v \in \mathbb{S}^{d-1}$ be a fixed point on a sphere, and $X_1, X_2, \cdots, X_n \sim Unif(\mathbb{S}^{d-1})$ which are i.i.d.

How large $n$ can guarantee that, with probability $1-\delta$, $\exists i$ such that $\|v-X_i \|\leq \epsilon$?

I think sufficiently many samples could do this, but I don't know how to calculate.

What about the case when $X_1, X_2, \cdots, X_n \sim N(0,1)$?

Answer 1

Note that if $d(u,x)$ is the distance between $u$ and $x$, then $$ \mathsf{P}\!\left(\bigcap_{i=1}^n\{d(u,X_i)>\epsilon\}\right)=[\mathsf{P}(d(u,X_1)>\epsilon)]^n=\left[\frac{|\{x\in \mathbb{S}^{d-1}:d(u,x)>\epsilon\}|}{|\mathbb{S}^{d-1}|}\right]^n. $$ (Here $|\cdot|$ denotes area.)