A Simple Way to Prove the Characterization of Differentiable Quasiconvex Functions ()
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
![](https://www.scirp.org/html/htmlimages\12-7401935x\965b893f-6a7a-42aa-bd81-973e1d80c46c.png)
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
![](https://www.scirp.org/html/htmlimages\12-7401935x\d1946529-1452-4e73-be61-4547aa1ad7e2.png)
i.e., in a more symmetric way,
![](https://www.scirp.org/html/htmlimages\12-7401935x\833d6522-ed0a-42ce-8802-dba8ce4eea81.png)
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
![](https://www.scirp.org/html/htmlimages\12-7401935x\e1709012-9dae-4c70-9a81-29f5ad772433.png)
Proof. The quasiconvexity of
is equivalent to the implication
![](https://www.scirp.org/html/htmlimages\12-7401935x\47d460a7-72c2-4a93-b9b0-ae7b961953b5.png)
By setting
we have
The thesis follows by noting that
and the logical implication
,
are equivalent to
and
,
respectively. ![](https://www.scirp.org/html/htmlimages\12-7401935x\ef87ba80-ed23-43cd-a05c-4790cb7e1e13.png)
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 ![](https://www.scirp.org/html/htmlimages\12-7401935x\2ed2a994-0d15-4905-868a-a1459b01e69a.png)
such that
Let
the continuity of
implies the existence of
such that
and ![](https://www.scirp.org/html/htmlimages\12-7401935x\e16b1054-2601-4693-ae41-215595f7ac24.png)
The mean value theorem applied to the interval
implies the existence of
such that
Consequently, we have
with
and this contradicts (2). ![](https://www.scirp.org/html/htmlimages\12-7401935x\ff686fa3-2cbd-4e89-9500-f30a43549ed5.png)
Proof of Theorem 1.
It is sufficient to note that if
is differentiable on the open and convex set
then we have, with ![](https://www.scirp.org/html/htmlimages\12-7401935x\a1905e35-c400-44e7-b384-4987d3e1a757.png)
![](https://www.scirp.org/html/htmlimages\12-7401935x\1d0f82d6-de65-4c0b-a785-4b66b99d6e6f.png)
Therefore, on the ground of Lemmas 1 and 2, if we put
we have
i.e. ![](https://www.scirp.org/html/htmlimages\12-7401935x\a7835791-172f-4c47-9b39-abe39a9916a3.png)
![](https://www.scirp.org/html/htmlimages\12-7401935x\e321ef9b-953b-47f1-9223-b4890003d343.png)
![](https://www.scirp.org/html/htmlimages\12-7401935x\0ff3fc18-a8ea-41d1-97e3-baea9034a372.png)
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
![](https://www.scirp.org/html/htmlimages\12-7401935x\dc091c93-6f82-4f0d-973e-20c2dc8b4ef1.png)
By the mean value theorem there exists a number
such that
![](https://www.scirp.org/html/htmlimages\12-7401935x\4570faa1-26fd-444f-95fc-e6831fb22161.png)
Dividing by
and letting
we have ![](https://www.scirp.org/html/htmlimages\12-7401935x\a431aa66-132d-4318-9ff5-e79a852d8996.png)
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. ![](https://www.scirp.org/html/htmlimages\12-7401935x\aff7d550-a98b-4d32-8d71-f236be37cfda.png)