For any given $x \in \mathbb{R}^n$, let $\nabla{x} \in \mathbb{R}^{n \choose 2}$ be the vector whose {i,j}-th entry is $|x_i-x_j|$. I think the following claim is true.

Claim. If $f, g \in \mathbb{R}^n$ are vectors with zero mean, i.e., $$\sum_{i=1}^n f_i = \sum_{i=1}^n{g_i}=0$$ and the angle between them is at most $\frac{\pi}{2}$, then $$\mbox{dist} (\nabla{f},\nabla{g}) \ge \mbox{dist}(f,g).$$

If anybody has any idea about how to prove this, please share it with me. Thanks.