The convexity of the solution set of a pseudoconvex inequality

Abstract

For a pseudoconvex function f on a nonempty convex set C in a real normed vector space X , we present several equivalent conditions for the convexity of the set C x ≔ { c ∈ C : f ( x ) ≤ f ( c ) } for x ∈ C . These conditions turn out to be very useful in characterizing the solution set of a pseudoconvex minimization problem of f over C x and the pseudolinearity of a Gâteux differentiable function f . We hence extend several existing results about characterizations of the solutions to a convex program and a pseudolinear program.