 smoothness is 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, convex and there exists $i_0\in I$ such that $f_{i_0}$ is uniformly smooth. Then $p_\mu({\bf f},{\boldsymbol \lambda})$ is uniformly 