A porosity result in convex minimization

Abstract

We study the minimization problem $f\left(x\right)\to min$, $x\in C$, where $f$ belongs to a complete metric space $ℳ$ 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 $ℱ$ be the set of all $f\in ℳ$ 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 we show that the set $ℱ$ contains an everywhere dense $G_{\delta }$ subset of $ℳ$. In this paper, we show that the complement $ℳ\backslash ℱ$ is not only of the first Baire category but also a $\sigma$-porous set.