﻿ A Simple Way to Prove the Characterization of Differentiable Quasiconvex Functions

Applied Mathematics
Vol.5 No.8(2014), Article ID:45702,3 pages DOI:10.4236/am.2014.58114

A Simple Way to Prove the Characterization of Differentiable Quasiconvex Functions

Giorgio Giorgi

Faculty of Economics, University of Pavia, Pavia, Italy

Email: ggiorgi@eco.unipv.it  Received 23 January 2014; revised 23 February 2014; accepted 2 March 2014

ABSTRACT

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

Keywords:Quasiconvex Functions, Generalized Convexity 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  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  . 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  ; 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.  - ). 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  and is given also by Crouzeix  , 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  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. References

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