I am having difficulty solving the following optimisation problem.
$$ \begin{array}{ll} \underset {x, y \in \Bbb R} {\text{maximize}} & L(x,y) := y \sum\limits_{k=0}^{N-1} \left| \cos\left(x+2\pi\frac{nk}{N} \right) \right| \\ \text{subject to} & y \left|\cos\left(x+2\pi\frac{nk}{N}\right)\right|-1 \le 0, \qquad k \in \{ 0, 1, 2, \dots, N-1 \} \end{array} $$
where $n, N \in \Bbb N^+$. Looking at numerical solutions it appears the optimum value of $x$ is $\pi / N$, but I don't know how to prove this.