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 subset of X is a nonempty convex set, we show that for every lower semicontinuous function h : X -> R boolean OR {+infinity} that satisfies sup(delta>0) inf(x is an element of E+delta BX)h(x) > h(a) (B-X being the unitary open ball in X), there exists (x) over bar is an element of [a, E] such that h(a) >= h((x) over bar) and (x) over bar is a strict minimizer of h on the drop [(x) over bar, E].