Let $\mathbf{x}=[x_i]_{i=1}^d, \mathbf{y}=[y_i]_{i=1}^d$ be two vectors in $R^d$.
Is it possible to find a lower bound $l\leq \|x-y\|$ and an upper bound $u\geq\|x-y\|$ as a function of $\bar{\mathbf{x}}, \bar{\mathbf{y}}, \sigma({\mathbf x)}$ and $\sigma({\mathbf y})$ ,
where ${\bar v} = \frac{1}{d}\sum_{i=1}^d{v_i}$ and $\sigma(v)=\sqrt{\frac{1}{d}\sum_{i=1}^d{(v_i-\bar v)^2}}$ (the mean and the standard the deviation of $\mathbf{v}$)?