 convexity 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 and $\epsilon_i$-convex. Then $p_\mu({\bf f},{\boldsymbol \lambda})$ is $\epsilon$-convex where $\epsilon=r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})$. In particular, if there exists $i_0\in I$ such that $f_{i_0}$ is $\epsilon_{i_