l{\alpha} \in
\mathbb{R}^{d}_{+}\)</span> is given, for all <span class="math inline">\(\boldsymbol{w} \in \mathbb{S}_d\)</span>, by <span class="math display">\[\begin{align*}
h_{\boldsymbol{W}}(\boldsymbol{w})=\frac{\Gamma(\bar{\alpha}+\rho)}{d\rho^{d-1}\prod_{i=1}^d\Gamma(\alpha_i)}
\bigl\langle\{\boldsymbol{c}(\boldsymbol{\alpha},\rho)\}^{1/\rho},\boldsymbol{w}^{1/\rho}\bigr\rangle^{-\rho-\bar{\alpha}}\prod_{i=1}^{d}
\{c(\alpha_i,\rho)\}^{\alpha_i/\rho}w_i^{\alpha_i/\rho-1}.
\end{align*}\]</span> where 