A porosity result in convex minimization (Q2495008)

Summary: We study the minimization problem \(f(x)\to\min\), \(x\in C\), 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 \(i\to\infty\) to the solution over the set \(C\). In our recent work [Commun. Appl. Anal. 5, No.~4, 535--545 (2001; Zbl 1085.49507)] we have shown 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.