# $2\sum_{i,j} |x_i-y_j| \geq \sum_{i,j} (|x_i-x_j| + |y_i-y_j|)$?

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:

1. For small $$n$$, I randomly generated $$x_i, y_i$$ many times and checked that the inequality holds.

2. 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 $$\sum_{i,j=1}^n (|x_i-x_j| + |y_i-y_j|) \leq 4 \sum_{i,j=1}^n |x_i-y_j|.$$

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$$2 \sum_{0
$$\sum_{i,j=1\leq n}(x_i-y_j)-\sum_{i,j=1\leq n}(x_j-y_i) \leq \sum_{i,j=1\leq n}|x_i-y_j| +\sum_{i,j=1\leq n}|x_j-y_i| = 2\sum_{i,j=1\leq n}|x_j-y_i|$$