An optimization problem
The problem as described is not convex, due to the nonconvexity of the constraint set. However, if you were to permit a relaxation, we could write your problem as
$\begin{array}{ll}
\max &amp; \sum_{p\in P} t_p\\
\text{subject to }&amp; t_p \geq \langle x, \|p\| p \rangle\\
&amp; t_p \geq -\langle x, \|p\| p\rangle ,\\
&amp;  \|x\|_2 \leq 1. 
\end{array}$
This is a second-order cone program, which can be solved by standard methods.
It is probably not trivial to recover a solution to your original problem from this relaxation (for instance, Goemans-Williamson wrote a breakthrough paper describing the rounding algorithm for a similar relaxation). But maybe this is helpful in some way.
edit -- As pointed out by Neal Young, there is an error in the above answer. The question in the original post is a maximization of absolute value function, which isn't a convex objective; so, my reduction is incorrect. I am leaving the incorrect answer here in case someone finds any information in it useful, but please see Neal Young's comments below for more comprehensive (and correct) observations about the problem.