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 v-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.