The domination property for efficiency and Bishop–Phelps theorem in locally convex spaces

We introduce the notion of Θ -closedness for any bounded convex set Θ and investigate the relationship between Θ -closedness and local closedness. Moreover we study some properties of σ -convex sets. By using Θ -closedness, local closedness and σ -convexity, we give two main theorems on the domination property and the existence of efficient points in locally convex spaces. From these two main theorems we deduce a number of corollaries, which improve the related known results. As an application of the main results, we obtain several support point theorems in locally convex spaces, which generalize the famous Bishop–Phelps theorem.