Abstract. In this paper, our aim is to discuss the density of proper efficient points. As an interesting application of the results in this paper, we want to prove a density theorem of Arrow, Barankin, and Blackwell.In [1], Luc introduced a new concept of the proper efficient point for a set. Using some results of recession cone, Luc established efficiency conditions, especially proper efficiency and domination properties ( [1, 2]). The present paper is devoted to the study of the density of proper efficient points. In detail, the set of proper efficient points for a set is dense in the set of efficient points. As an interesting application of the results in this paper, we prove a density theorem of Arrow, Barankin, and Blackwell ( [3,4]).First let us recall some notations: Throughout the paper, E is a separated locally convex topological linear space and E * its topological dual. U (0) denotes the family of balanced open convex neighbourhoods of the origin in E. For A ⊂ E, cone(A), cl(A), and int(A) denote the generated cone, the closure, and the interior of A, respectively.Let C ⊂ E be a convex cone, and let A be a nonempty subset of E. We say that x ∈ A is an efficient point of A with respect to C if there exists y ∈ A, such thatWe denote by E(A, C) the set of all efficient points of A (with respect to C). We say that x ∈ A is a proper efficient point of A with respect to C if there exists a closed convex cone K = E such that C\{0} ⊂ int(K) and x ∈ E(A, K).The set of proper efficient points of A is denoted by Prop E(A, C). It is obvious that the set of proper efficient points of A is contained in the set of efficient points,but the converse is not generally true.If C is a convex cone, the convex set B ⊂ C is said to be a base of C if 0 / ∈ cl(B) and C = cone(B) = {tB : t ≥ 0} = {tb : t ≥ 0, b ∈ B}.A cone with base must be pointed.