The convexity of the solution set of a pseudoconvex inequality

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 {colon equals} {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.