We wish to maximize the minimum of a weighted sum of cosines in the plane, when the frequency components are on the unit circle. Formally: $$\max_{\{ a_i,\theta_i,\phi_i \}_{i=1}^{N} } \min_{(x,y) \in \mathbb{R}^2} \sum_{k=1}^N a_k \cos(\cos(\theta_k)x+ \sin(\theta_k)y + \phi_k) $$ $$ \mathrm{such} \,\,\, \mathrm{that}: \forall i: \theta_i\in[0,\pi),\phi_i\in[0,\pi), a_i\in \mathbb{R}, \sum_{k=1}^{N}a_k^2=1, \forall j\neq i: \theta_i \neq \theta_j $$
Prove (or disprove): for any $N\geq3$ a maximizer of this objective is the symmetric hexagonal solution
$$ \theta_1=0,\theta_2=\pi/3,\theta_3=2\pi/3 ,$$
with $\forall i=1,2,3: a_i=\frac{1}{\sqrt3}$ and $\phi_i=0$, and $\forall i>3: a_i=0$.
Background: I believe this problem is strongly connected to my previous question Maximal minimum for a sum of two (or more) cosines .
Thanks in advance!
