Existence of minimizers on drops (Q2848189)

The purpose of this paper is to show that there exist minimizers in some particular contexts where the mentioned assumptions do not necessarily hold. Namely, for a given lower semicontinuous function \(h: X \rightarrow R \cup \{ + \infty \}\) defined on a Banach space \(X\), a nonempty convex set \(E \subset X\) (closed or open) and a point \(a \in X\) such that \(h(a)<h(x)\) for all \(x\) in a neighborhood \(V\) of E are considered. The authors prove that there exists \( \overline{x} \in [a,E]\backslash V\) such that \(h(\overline{x}) \leq h(a)\) and \(h(\overline{x})< h(x)\) for all \(x\in [\overline{x},E]\backslash \{\overline{x}\}\) (as soon as the drop \([a,E]:=\{ ta+(1-t)y | t\in [0,1], y \in E\}\) is boundedly generated). In particular, from this result for some point \(\overline{x}\) in the drop \([a,E]\), the function \(h\) has its strict minimum at \(\overline{x}\) on every segment \([\overline{x},e]\) with \(e\in E\), follows. This is an existence result for a minimizer to be at a vertex of a truncated cone generated by the set \(E\).