Characterizations of the set containment with star-shaped constraints

In this paper, we first give a separation theorem for a closed star-shaped set at the origin and a point outside it in terms of separation by an upper semi-continuous and super-linear function, and also, we introduce a ν-star-shaped-conjugation. By using this facts, we present characterizations of the set containment with infinite star-shaped constraints defined by weak inequalities. Next, we give characterizations of the set containment with infinite evenly radiant constraints defined by strict or weak inequalities. Finally, we give a characterization of the set containment with an upper semi-continuous and radiant constraint, in a reverse star-shaped set, defined by a co-star-shaped constraint. These results have many applications in Mathematical Economics, in particular, in Utility Theory.