The convexity of the solution set of a pseudoconvex inequality

For a pseudoconvex function ff on a nonempty convex set CC in a real normed vector space XX, we present several equivalent conditions for the convexity of the set Cx≔{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 ff over CxCx and the pseudolinearity of a Gâteux differentiable function ff. We hence extend several existing results about characterizations of the solutions to a convex program and a pseudolinear program.