Why these two problems lead to same answers?

Question

Suppose these two problems:
Problem 1: $$\min_{X,P} \quad\max_{1\leq l\leq L-1} \quad {|\sum_{1\leq i\leq N_p}^{N_p}x_ie^{\frac{2\pi l}{N}p_i}| \over {\sum_{i=1}^{N_p} x_i^2}} \quad \equiv \quad \min_{X,P} \quad\max_{1\leq l\leq L-1} \quad {\sum_{1\leq i,j\leq N_p}^{N_p}x_ix_je^{\frac{2\pi l}{N}(p_i-p_j)} \over {(\sum_{i=1}^{N_p} x_i^2})^2} $$ Problem 2: $$\min_{P} \quad\max_{1\leq l\leq L-1} \quad |\sum_{1\leq i\leq N_p}^{N_p}e^{\frac{2\pi l}{N}p_i}| \quad \equiv \quad \min_{P} \quad\max_{1\leq l\leq L-1} \quad \sum_{1\leq i,j\leq N_p}^{N_p}e^{\frac{2\pi l}{N}(p_i-p_j)} $$ Where $P=\{p_1,p_2,...,p_{N_p}\},\quad 1 \leq p_i\leq N , \quad p_i \in \mathbb{N}, \quad p_i \ne p_j, \quad X \in \mathbb{R}^{N_p}, \quad l \in \mathbb{N}$
As a matter of fact, Problem 2 is Problem 1 when we set $X$ as a vector with same entries. After lots of simulations I've noticed that these two optimization problems lead to same solution of $P^*$. I'm looking for a proof or at least some justifications to be able to write a proof for it?
note that the optimal $X$ for Problem 1 is not $X$ with equal entries!