 dominant w.r.t. $p_\mu$)*** Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. If there exists $i_0\in I$ such that ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}f_{i_0}\neq\varnothing$, then ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}p_\mu({\bf f},{\boldsymbol \lambda})\neq\varnothing$. If there exists $i_0\in I$ such that ${\ensuremath{\operatorname{dom}}}f_{i_0}={\ensuremath{\mathcal H}}$, then ${\ensuremath{\operatorname{dom}}}p_\mu({\bf f},{\boldsymbol \lambda})={\ensuremath{\mathcal H}}$.  For each $i\in I$, we set $A_i=\partial f_i$. Suppose that for some $i_0\in I$, ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}f_{i_0}\neq\varnothing$