Let $f,g\in\mathbb{R}^n$ be two vectors with zero mean (i.e. $\sum_{i}{f_i}=\sum_{i}{g_i}=0$). Let $\nabla{f}\in\mathbb{R}^{{n}\choose{2}}$ be the vector that its ${\{i,j\}}$th entry is $|f_i-f_j|$, and $\nabla{g}\in\mathbb{R}^{{n}\choose{2}}$ be the vector that its ${\{i,j\}}$th entry is $|g_i-g_j|$.

Question: Can we prove $dist(\nabla{f},\nabla{g}) \ge dist(f,g)$?