Let $x_1, \ldots, x_n$ and $y_1, \ldots, y_n$ be real numbers.
Question: Is $$2\sum_{i,j=1}^n |x_i-y_j| \geq \sum_{i,j=1}^n (|x_i-x_j| + |y_i-y_j|)$$ true?
Motivation: This is motivated from a physical problem. If the above inequality holds, then one can obtain that certain terms are suppressed exponentially.
My trial:
For small $n$, I randomly generated $x_i, y_i$ many times and checked that the inequality holds.
If the prefactor in LHS is 4, then I can prove the inequality as follows. For $k=1, \ldots, n$, we have $$|x_i-x_j| \leq |x_i-y_k| + |y_k-x_j|$$ and analogously $$|y_i-y_j| \leq |y_i-x_k| + |x_k - y_j|.$$ Averaging the above inequalities over all $k$ and summing over $i,j$ gives $$\sum_{i,j=1}^n (|x_i-x_j| + |y_i-y_j|) \leq 4 \sum_{i,j=1}^n |x_i-y_j|.$$