First let me define *Difference multiset* for a set of integers
$$P=\{p_1,p_2, \dots,p_K\} ,\quad p_i \in\{1,2,\dots,N\},\quad p_i\ne p_j
$$
as below:
$$
D = \{p_i-p_j \mod N ,\quad i \ne j\}
$$
I know that the minimum of $f(x)$ is same for all $P$'s having same difference multiset (homometric $P$'s) & and also the optimal $x$ (minimizing $f$) is same for all of them up to permutation of elements, (also for all homometric $P$'s same $l$ is the inner maximizer) , where $f(x)$ is a real function which is defined as:

$$\large f(x) = \max_{1 \leq l \leq N-1} {\sum_{j=1}^K \sum_{k=1}^K x_j x_k e^{\frac{i2\pi l(p_j-p_k)}N} \over \left( \sum_{i=1}^K x_i\right)^2 } = \max_{1 \leq l \leq N-1} { \sum_{i=1}^K x_i^2 + \sum_{j,k} x_j x_k cos(\frac{2\pi l(p_j-p_k)}N) \over \left( \sum_{i=1}^K x_i\right)^2 } $$ $x_i$'s are positive variables

I achieved this result from simulations. I'm looking for a proof or even a justification which helps me prove it.

Let me give you an example of my simulations if it helps:

suppose $(N,K)= (6, 4)$

$$ P = \{1,2,3,5\} \Rightarrow D = \{1,1,2,2,2,3,3,4,4,4,5,5\} $$ minimizing $f(x)$ with $p_i$'s being members of $P$, led to this $x=(4,5,4,4)$

and for $$ P' = \{1,2,4,6\} \Rightarrow D' = \{1,1,2,2,2,3,3,4,4,4,5,5\} = D $$ minimizing $f(x)$ with $p_i$'s being members of $P'$, led to this $x=(5,4,4,4)$

Also note that $f(x) $ is independent of $||x||_2$ and is just function of angle of vector $x$.

I've also asked this on ME

whatyou want to show) in the title. That could give people an idea of what the question is about, possibly prompting them to read the question. – Ricardo Andrade Jul 23 '13 at 8:47