A Simple Way to Prove the Characterization of Differentiable Quasiconvex Functions
Giorgio Giorgi
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Abstract

We give a short and easy proof of the characterization of differentiable quasiconvex functions.

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Giorgi, G. (2014) A Simple Way to Prove the Characterization of Differentiable Quasiconvex Functions. Applied Mathematics, 5, 1226-1228. doi: 10.4236/am.2014.58114.

1. Introduction

Quasiconvex functions play an important role in several branches of applied mathematics (e.g. mathematical programming, minimax theory, games theory, etc.) and of economic analysis (production theory, utility theory, etc.). De Finetti [1] was one of the first mathematicians to define quasiconvex functions as those functions, being a convex set, having convex lower level sets, i.e. the set

is convex for every.

De Finetti did not name this class of functions: the term “quasiconvex (quasiconcave) function” was given subsequently by Fenchel [2] . It is well-known that the above characterization is equivalent to

i.e., in a more symmetric way,

When f is differentiable on the open convex set we have the following characterization of a quasiconvex function.

Theorem 1. Let be differentiable on the open convex set Then f is quasiconvex on X if and only if

(1)

Theorem 1 was given by Arrow and Enthoven [3] ; however, these authors prove, in a short and easy way, only the necessary part of the theorem, but not the converse property, whose proof is indeed presented in a quite intricate way by several authors (see, e.g. [4] -[11] ). Here we present an easy proof of Theorem 1, by exploiting some results on quasiconvexity of functions of one variable, results therefore suitable for geometrical illustrations. We need two lemmas.

Lemma 1. Let, a convex set. Then f is quasiconvex on X if and only if the restriction of f on each line segment contained in X is a quasiconvex function, i.e. if and only if the function is quasiconvex on the interval

Proof. The quasiconvexity of is equivalent to the implication

By setting we have The thesis follows by noting that and the logical implication, are equivalent to and, respectively.

The next lemma is proved in Cambini and Martein [12] and is given also by Crouzeix [13] , without proof.

Lemma 2. Let be differentiable on the interval; then is quasiconvex on I if and only if

(2)

Proof. Let such that and The quasiconvexity of implies so that is locally non-increasing (locally non-decreasing) at t1 and consequently (2) holds. Assume now that (2) holds. If is not quasiconvex, there exist with

such that Let the continuity of

implies the existence of such that and

The mean value theorem applied to the interval implies the existence of

such that Consequently, we have with and this contradicts (2).

Proof of Theorem 1.

It is sufficient to note that if is differentiable on the open and convex set then we have, with

Therefore, on the ground of Lemmas 1 and 2, if we put we have i.e.

Finally, we point out that the proof of Ponstein [10] can be shortened as follows:

1) Let be quasiconvex (and differentiable) on the open and convex set i.e. let

By the mean value theorem there exists a number such that

Dividing by and letting we have

2) Assume conversely that (1) holds and that for with there exists a point between and with Then there exists near (e.g. between and) also a point with and Indeed, if for all x between and it would hold or then we would have but not But, being, this implies, in contradiction with the inequality previously obtained.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] De Finetti, B. (1949) Sulle Stratificazioni Convesse. Annali di Matematica Pura ed Applicata, 30, 173-183.http://dx.doi.org/10.1007/BF02415006 [2] Fenchel, W. (1953) Convex Cones, Sets and Functions. Lecture Notes, Princeton University, Princeton. [3] Arrow, K.J. and Enthoven, A.C. (1961) Quasi-Concave Programming. Econometrica, 29, 779-800.http://dx.doi.org/10.2307/1911819 [4] Avriel, M. (1976) Nonlinear Programming: Analysis and Methods. Prentice-Hall, Englewood Cliffs. [5] Avriel, M., Diewert, W.E., Schaible, S. and Zang, I. (1988) Generalized Concavity. Plenum Press, New York.http://dx.doi.org/10.1007/978-1-4684-7600-2 [6] Bazaraa, M.S. and Shetty, C.M. (1976) Foundations of Optimization. Springer Verlag, Berlin.http://dx.doi.org/10.1007/978-3-642-48294-6 [7] Bazaraa, M.S., Sherali, H.D. and Shetty, C.M. (1993) Nonlinear Programming. John Wiley & Sons, New York. [8] Kemp, M.C. and Kimura, Y. (1978) Introduction to Mathematical Economics. Springer Verlag, New York.http://dx.doi.org/10.1007/978-1-4612-6278-7 [9] Mangasarian, O.L. (1969) Nonlinear Programming. McGraw-Hill, New York. [10] Ponstein, J. (1967) Seven Kinds of Convexity. SIAM Review, 9, 115-119. http://dx.doi.org/10.1137/1009007 [11] Simon, C.P. and Blume, L. (1994) Mathematics for Economists. W. W. Norton & Co., New York. [12] Cambini, A. and Martein, L. (2009) Generalized Convexity and Optimization. Springer, Berlin. [13] Crouzeix, J.-P. (2005) Criteria for Generalized Convexity and Generalized Monotonicity in the Differentiable Case. In: Hadjisavvas, N., Komlosi, S. and Schaible, S., Eds., Handbook of Generalized Convexity and Generalized Monotonicity, Springer, New York, 89-119. http://dx.doi.org/10.1007/0-387-23393-8_2

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