Characterizations of the set containment with star-shaped constraints

Document Type : Research Paper


1 Department of Mathematics, Islamic Azad University, Kerman Branch, Kerman, Iran

2 Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran


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 $\nu$-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.


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Volume 12, Issue 1
May 2021
Pages 790-811
  • Receive Date: 03 February 2018
  • Revise Date: 19 November 2018
  • Accept Date: 25 November 2018
  • First Publish Date: 01 May 2021