Structure of the efficient point set

Abstract

Let C C be a nontrivial cone and X X be a set in the n n -dimensional Euclidean space. Denote by E ( X | C ) {\text {E}}(X|C) the set of all efficient points of X X with respect to C C . It will be proven that under some adequate assumptions E ( X | C ) {\text {E}}(X|C) is homeomorphic to a simplex while n = 2 n = 2 , and for n > 2 n > 2 it is a contractible set. Furthermore, the set of all weak efficient points of X X with respect to C C is arcwise connected and its local contractibility is equivalent to being a retract of X X . The results presented in this study cover all topological properties of the efficient point set which have been obtained by Peleg and Morozov for the case when C C is the nonnegative orthant.