Existence of minimizers on drops

Abstract

For a boundedly generated drop [a,E] (a property which holds, for instance, whenever E is bounded), where a belongs to a real Banach space X and E ⊂ X is a nonempty convex set, we show that for every lower semicontinuous function h : X −→ R ∪ {+∞} that satisfies supδ>0 infx∈E+δBX h(x) > h(a) (BX being the unitary open ball in X), there exists ¯x ∈ [a,E] such that h(a) ≥ h(¯x) and ¯x is a strict minimizer of h on the drop [¯x,E].