We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers with the $x_i$ pairwise distinct. We use the notation $\alpha_*$, $h_\alpha$, $K$, $a$, $\mathcal{S}_*$, $J_*$, $(\mid)_{\mathcal{H}}$ as defined previously.
Deduce that there exists $\alpha_* \in \mathbf{R}^p$ such that $\mathcal{S}_* = \{h_{\alpha_*}\}$ and calculate the value of $J_*$ in terms of $K$ and $a$.