We fix $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$. Show that if $V_{n}(\boldsymbol{x}) = \frac{1}{n} \sum_{i=1}^{n} |\boldsymbol{x}_{i} - \overline{\boldsymbol{x}}|^{2}$ and $V_{n}(\boldsymbol{y}) = \frac{1}{n} \sum_{i=1}^{n} |\boldsymbol{y}_{i} - \overline{\boldsymbol{y}}|^{2}$ then $$\delta(\boldsymbol{x}, \boldsymbol{y})^{2} = nV_{n}(\boldsymbol{x}) + nV_{n}(\boldsymbol{y}) - 2\sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle Z(\boldsymbol{x}, \boldsymbol{y}), R \rangle$$ where $Z(\boldsymbol{x}, \boldsymbol{y})$ is a matrix that we will specify.
We fix $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$. Show that if $V_{n}(\boldsymbol{x}) = \frac{1}{n} \sum_{i=1}^{n} |\boldsymbol{x}_{i} - \overline{\boldsymbol{x}}|^{2}$ and $V_{n}(\boldsymbol{y}) = \frac{1}{n} \sum_{i=1}^{n} |\boldsymbol{y}_{i} - \overline{\boldsymbol{y}}|^{2}$ then
$$\delta(\boldsymbol{x}, \boldsymbol{y})^{2} = nV_{n}(\boldsymbol{x}) + nV_{n}(\boldsymbol{y}) - 2\sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle Z(\boldsymbol{x}, \boldsymbol{y}), R \rangle$$
where $Z(\boldsymbol{x}, \boldsymbol{y})$ is a matrix that we will specify.