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 := {c is an element of C : f(x) <= f(c)} for x is an element of C. These conditions turn out to be very useful in characterizing the solution set of a pseucloconvex minimization problem of f over C-x and the pseudolinearity of a Gateux differentiable function f. We hence extend several existing results about characterizations of the solutions to a convex program and a pseudolinear program. (C) 2007 Elsevier Ltd. All rights reserved.