We study the minimization problem $f(x) \rightarrow \min$ , $x \in C$ ,
where $f$ belongs to a complete metric space $\mathcal{M}$ of
convex functions and the set $C$ is a countable intersection of a
decreasing sequence of closed convex sets $C_i$ in a reflexive
Banach space. Let $\mathcal{F}$
be the set of all $f \in \mathcal{M}$
for which the solutions of the minimization problem
over the set $C_i$ converge strongly as $i \rightarrow \infty$ to the solution over the set $C$ . In our recent work we show that
the set $\mathcal{F}$ contains an everywhere dense $G_{\delta}$ subset of $\mathcal{M}$ . In this paper, we show that the
complement $\mathcal{M} \setminus \mathcal{F}$ is not only of the
first Baire category but also a $\sigma$ -porous set.