For $u$ a unit vector and $x_1\ldots,x_n$ a set of fixed points in $R^d$, consider $f(u)=\sum_{i=1}^n \frac{\langle x_i,u\rangle_+}{\langle x_i,u\rangle_-}$, where $a_+$ and $a_-$ are the corresponding positive and negative parts of a real $a$. Is there a general way of finding $\sup_u f(u)$?
So far, for $d=2$ I've managed to solve the problem in terms of polar coordinates (the optimal comes in terms of the angles of the given points, as I expected). However, I'm not an expert in this subject, so I was wondering if there's a known general solution.
Thank you in advance!
