Assume $n$-dimensional unit Ball $B^n(0,1)$ in an Euclidean space. I have a set of points, say $S$ to discretize it. How can I compute the maximum error in $l_2$? To be more precise, every point $s$ in $S$ has its elements drawn from $\{-1,0,1\}$ and it is then normalized (i.e. all the $3^n$ combinations of $\{-1,0,1\}$ normalized).
Edit: Doing some homework, I want to $$\max_{\|x\|_2=1}\min_{s_i\in S}\|x-s_i\|_2.$$ Since $\|x-s_i\|_2^2=2-2\cos{\theta_i}$ where $\theta_i$ is the acute angle between $x$ and $s_i$ and since they are unit-length vectors we have that $\cos{\theta_i}=s_i^\top x$ then the problem is equivalent to $$\min_{\|x\|_2=1} \max_{s_i\in S} s_i^\top x= \min_{\|x\|_2=1} \|S^\top x\|_\infty$$