Structure of the efficient point set

Abstract:

Let $C$ be a nontrivial cone and $X$ be a set in the $n$-dimensional Euclidean space. Denote by ${\text {E}}(X|C)$ the set of all efficient points of $X$ with respect to $C$. It will be proven that under some adequate assumptions ${\text {E}}(X|C)$ is homeomorphic to a simplex while $n = 2$, and for $n > 2$ it is a contractible set. Furthermore, the set of all weak efficient points of $X$ with respect to $C$ is arcwise connected and its local contractibility is equivalent to being a retract of $X$. 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 $C$ is the nonnegative orthant.