Suppose we have two vectors $x = (x_1, \dots, x_n)^T, y = (y_1, \dots, y_n)^T \in \mathbb{R}^n$ where the suberscript denote the components. Let $\bar{x} = \frac{\sum_{j=1}^n x_j} {n} \mathbf 1$ where $\mathbf 1$ is the all $1$ vector. I want to find some upper bound for $\|x-y\|_2^2$ in terms of $\|\bar{x} - \bar{y}\|_2^2$. \begin{align*} \|x-y\|_2^2 = (x_1 - y_1)^2 + \dots + (x_n - y_n)^2 \\ \|\bar{x} - \bar{y} \|_2^2 = (1/n) \left( \sum_{j=1}^n x_j - \sum_{i=1}^n y_i \right)^2. \end{align*}
As pointed by the comment, we need to put the assumption $\bar{x} \neq \bar{y}$. But I am not sure whether there exists some sensible constant $C$ such that \begin{align*} \|x-y\|_2^2 \le C \| \bar{x} - \bar{y} \|_2^2. \end{align*} Any suggestions? Thanks.
No such estimate can exist. The essential reason is what Kavi Rama Murthy said in above comment:
For continuity reasons it does not help to add the assumption $\bar{x} \neq \bar{y}$. A concrete counter-example is
$$ x = (1, 0, 0, \ldots, 0)^T \, , \, y = (K, -K, 0, \ldots, 0)^T $$ for large $K$.