Given two arbitrary vectors: $\vec X$ and $\vec Y$, where $0<||\vec X||_2,||\vec Y||_2<1$, with any non-zero angle.
Find the optimal decomposition of $\vec X$ and $\vec Y$: $\vec X=\sum_{i=0}^m \vec x_i$, $\vec Y=\sum_{j=0}^n \vec y_j$, and $\sum_{i=0}^m||\vec x_i||_2=\sum_{j=0}^n||\vec y_j||_2=1$ (each $\vec x_i$ or $\vec y_j$ has a different angle), to maximize the sum of absolute values of inner products. $$\text{Maximize} \sum_{i=0}^m \sum_{j=0}^n |\vec x_i \vec y_j|\\ \text{subject to:} \vec X=\sum_{i=0}^m \vec x_i, \vec Y=\sum_{j=0}^n \vec y_j, \sum_{i=0}^m||\vec x_i||_2=\sum_{j=0}^n||\vec y_j||_2=1$$
where m and n are also variables. I want to figure out that in the optimal decomposition, whether it is correct that m=n=1? (If so, then $||\vec x_0||_2+||\vec x_1||_2=1$ and $\vec x_0+\vec x_1=\vec X$, thus the track forms an ellipse, the same for $\vec y_i$) How to prove it?
This problem is nontrival, please do me a favor with the proof or upvote it, thx!