Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-682212120210201Characterizations of the set containment with star-shaped constraints790811353310.22075/ijnaa.2018.13932.1727ENArianHedayatDepartment of Mathematics, Islamic Azad University, Kerman Branch, Kerman, IranHosseinMohebiDepartment of Mathematics, Shahid Bahonar University of Kerman, Kerman, IranJournal Article20180203In 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 <span>$nu$</span>-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.https://ijnaa.semnan.ac.ir/article_3533_309418436f44ce75503bcaea860db887.pdf