Abstract:
© Heldermann Verlag We provide a short proof of following theorem, due to Delbaen and Orihuela and independently, Pérez-Aros and Thibault. Let A be a nonempty closed and bounded convex subset of a Banach space (X, k · k) and let W be a nonempty weakly compact subset of (X, k · k). If we have x∗0 ∈ {x∗ ∈ X∗ : supa∈A x∗(a) < 0} and argmax(y∗|A) 6= ∅ for each y∗ ∈ {x∗ ∈ X∗ : supa∈A x∗(a) < 0 and supw∈W |(x∗ − x∗0)(w)| < 1}, then A is weakly compact.