I wanted to solve the following optimization problem, where $\delta_{k,n}$ is a Kronecker delta: \begin{eqnarray*} ~\\ ~\textrm{Maximize}~~~\left|\sum\limits_{n=1}^N a_n\right|^2-\max\limits_{1\leq k\leq N}\left|\sum\limits_{n=1}^N(-1)^{\delta_{k,n}}a_n\right|^2\\ \textrm{subject to}~~\sum\limits_{n=1}^{N}{{{\left|a_n\right|}^{2}}}=1,~~a_n\in\mathbb{C}, n=1,\ldots N.\\ \end{eqnarray*}
I suspect that the maximum is reached for a uniform distribution of $a_n\in\mathbb{C}$, but I do not know where to start to prove it. For example, if ${{a}_{n}}=\frac{1}{\sqrt{N}}~\forall n$, then $\left|\sum_{n=1}^N a_n\right|^2=N$, $\max\limits_{1\leq k\leq N}\left|\sum_{n=1}^N(-1)^{\delta_{k,n}}a_n\right|^2=\frac{(N-2)^2}{N}$, and $\left|\sum_{n=1}^N a_n\right|^2-\max\limits_{1\leq k\leq N}\left|\sum_{n=1}^N(-1)^{\delta_{k,n}}a_n\right|^2=\frac{4(N-1)}{N}$. Is this really a maximum if $a_n\in\mathbb{C}$? Any help is much appreciated. Any idea where I should start? Thank you.