\] 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 $f_{i_0}$ is essentially smooth and there exists $i_0\in I$ such that $f_{i_0}$ is essentially strictly convex, then $p_\mu({\bf f},{\boldsymbol \lambda})$ is Legendre. In particular, if there exists $i_0\in I$ such that $f_{i_0}$ is 