# Density theorems for generalized Henig proper efficiency

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## Abstract

We develop a new, simple technique of proof for density theorems (i.e.,for the sufficient conditions to guarantee that the proper efficient points of a set are dense in the efficient frontier) in an ordered topological vector space. The results are the following: (i) the set of proper efficient points of any compact set*Q* is dense in the set of efficient points with respect to the original topology of the space whenever the ordering cone*K* is weakly closed and admits strictly positive functionals; moreover, if*K* is not weakly closed, then there exists a compact set for which the density statement fails; (ii) if*Q* is weakly compact, then we have only weak density, but if*K* has a closed bounded base, then we can assert the density with respect to the original topology, (iii) there exists a similar possibility to assert the strong density for weakly compact*Q* if additional restrictions are placed on*Q* instead of*K*. These three results are obtained in a unified way as corollaries of the same statement. In this paper, we use the concept of proper efficiency due to Henig. We extend his definition to the setting of a Hausdorff topological vector space.

### Key Words

Vector optimization positive proper efficiency Henig proper efficiency density theorem## Preview

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