Given $x_j\in\mathbb{R}^n$, $j=1,\ldots,p$, find

$$ \widehat{w} \in \arg\min_{\Vert w\Vert=1}\max_{1\le j\le p} |\langle w,x_j\rangle|. $$

I am also interested in the special case where we further constrain $w_i\ge0$ for all $i=1,\ldots,n$.

This is a convex program so it can be approximated numerically, but I am interested in an analytic / closed-form solution as a function of the $x_j$, i.e. $\widehat{w}=\widehat{w}(x_1,\ldots,x_p)$.

This problem seems distressingly simple but it has thwarted my attempts at a closed-form solution. Any ideas?

Given $x_j\in\mathbb{R}^n$, $j=1,\ldots,p$, find

$$ \widehat{w} \in \arg\min_{\Vert w\Vert=1}\max_{1\le j\le p} |\langle w,x_j\rangle|. $$

I am also interested in the special case where we further constrain $w_i\ge0$ for all $i=1,\ldots,n$.

This is a convex program so it can be approximated numerically, but I am interested in an analytic / closed-form solution as a function of the $x_j$, i.e. $\widehat{w}=\widehat{w}(x_1,\ldots,x_p)$.

This problem seems distressingly simple but it has thwarted my attempts at a closed-form solution. Any ideas?

Edit: As a result of Ilya's nice example, the following special case is still of interest to me: Take $j=1$ and assume $w\ge 0$ (i.e. all components of $w$ are nonnegative) -

$$ \widehat{w}(x) \in \arg\min_{\Vert w\Vert=1, w\ge 0}|\langle w,x\rangle|. $$