On approximate minima of a convex functional and lower semicontinuity of metric projections (Q803454)

Let X be a Banach space and let CC(X) be the class of all nonempty closed convex subsets of X. Given a function \(f:X\to {\mathbb{R}}\), a set C in CC(X), and a number \(\epsilon >0\), the set \(\{\) \(x\in C: f(x)\leq \inf f(C)+\epsilon \}\) is denoted by \(\epsilon\)-arg \(\min_ f(C)\). Under the assumpion that CC(X) is equipped with the Mosco topology, the authors deal with the continuity and upper semicontinuity of the function \[ C\mapsto \epsilon -\arg \quad \min_ f(C). \] Besides they investigate the connection between the lower semicontinuity of the metric projection and the existence of a metric selection for \(C\in CC(X)\).