Mathematical Notes, Vol. 61, No. ,~, 1997
B R I E F C O M M U N I C A T I O N S
Overlapping of Functions
F. V . G u s e i n o vKEY WORDS: convexification of functions, sublinear function, lower semicontinuity.
T h e
overlapping
of a function means that the graph of its convexification is contained in the convex hull of t h e g r a p h of the function. Here theconvezification
of a given function is the greatest lower semicontinuous function t h a t is not greater than the given function. Probably, it was Shoples and Shubik [1] who first discussed overlapping, at least in the context of mathematical economics (kernels and competitive balance). Shoples and Shubik pointed out that the overlapping condition can be used to regenerate a certain regularity lost when there are no concavity assumptions (the concavity of the utility function or the convexity of the preference pattern). Regularity is potentially useful in situations other than the direct context of kernels and competitive balances and seems to be essential for any analysis of the relaxation of convexity conditions.Some sufficient conditions for the overlapping of functions are contained in [1, Theorem 3]. Another sufficient condition was found in connection with the extension of variational problems [2, p. 278]. The author is not aware of other results concerning the overlapping of functions.
The present note is devoted to the study of the overlapping of functions. We obtain a necessary and sufficient overlapping condition for a lower semicontinuous function whose graph contains no lines (Theorem 1). Both results mentioned above are corollaries of Theorem 1. We also study the case of a function whose graph contains lines (Theorem 2).
As usual, by R = ]RU { - c o , oo} we denote the extended real axis, by ]R n real n-dimensional Euclidean space, by epi f , respectively
g r f ,
the epigraph, respectively, the graph, of a function f , and by f** the greatest lower semicontinuous function that is not greater t h a n f . T h e often used notation f** is suggested by the well known fact that f** is the second Legendre adjoint function of f . By int A andOA
we denote the interior and the boundary of a subset A C IR n . A point a of a convex set A is said to bean extreme point
if it can not be represented as a convex combination of two points of A different from a. The definition of anextreme ray
of a set A is similar. Aforward point
is an extreme point a such that there exists a plane of support passing through a and containing no points of A t h a t differ from a. Also recall t h a t a convex subset A' C A is said to be aface
if the endpoints of any segment of A belong to A' whenever some relatively interior point of this segment belongs to A'.For a subspace L C ]R n , by L • and
PL
we denote its orthogonal complement and the orthogonal projector onto L . A subset M0 of an affine subspace M C ]R" is called anenveloping subset
if its convex hull is the whole set M .D e f i n i t i o n . A function f : IR" --* 1R is called
overlapped
if for an arbitrary point x E ]R" there exist points x l , . . . , xm E IR n and positive numbers A1, . . . , Am (where A1 + -.- + Am = 1) such thatx = Aixi and
f**(x) =
Aif(xi),
(1)i=1 i=1
i.e., the g r a p h of f** is contained in the convex hull of the graph of f .
Translated from Matematicheskie Zametki, Vol. 61, No. 4, pp. 623-626, April, 1997. Original article submitted April 5, 1995.
It is easy to show t h a t the integer rn here can be b o u n d e d by n + 1.
T h e o r e m 1. Suppose that a function f : R n --+ R is lower semicontinuous and linearly bounded below, and epi f** does not contain iines. Then the function f is overlapped i f and only i f its g r a p h contains an unbounded subset of each nonvertical extreme ray of the epigraph of f**.
T h e condition on t h e function f that is formulated in the s t a t e m e n t of T h e o r e m 1 will be called the Condition.
P r o o f . Necessity. A s s u m e t h a t the condition of T h e o r e m 1 does not hold, i.e., there exists a nonvertical extreme ray ~ of an epigraph epi f** such t h a t the set ~ A gr f is b o u n d e d . By e denote the unit vector codirected w i t h t h e ray ~. Let ~ be the vertex of ~ a n d z0 a p o i n t of ~ such that the ray {(z0 -- e) q- Ae: a >_ 0) is contained in ~ a~d does not intersect the g r a p h of f :
{(z0 - + > 0} = 0 . (2)
T h e n t h e point zo = (xo,f**(xo)) has no representation of type (1). In fact, if such a representation exists, t h e n z0 is a relative interior point of t h e polyhedron I
P = conv{( , f( x k ) ) : k = 1 , . . . , m }
because Ak, k = 1 , . . . , rn, are strictly positive numbers. Since ~ is a face of e p i f * * , T h e o r e m 18.1 [3] implies t h a t t h e p o l y h e d r o n P is contained in 3. Combining this with (2), we o b t a i n
zk := ( x k , f ( x k ) ) e [~,zo--e], where k = l , . . . , m .
T h e n the p o l y h e d r o n P is also contained in the convex set [~, z0 - e], a n d t h u s z0 ~ P . This contradicts the inclusion z0 E ri P .
Su~ciency. T h e following s t a t e m e n t is an essential part of the proof of sufficiency.
P r o p o s i t i o n 1. Suppose that a function f : ~n __~ R satisfes the assumptions of Theorem 1. Then gr f D ex(epi f ) , i.e., f and f** have the same restrictions to the projection of the set of extreme points of epif** on R'*.
Further, t h e proof of Proposition I is based on the following s t a t e m e n t (which is of interest in itself). P r o p o s i t i o n 2. Suppose that a lower semicontinuous nonnegative function f : ~'~ --* R satisfies the Condition. Then the zero set of the function f** coincides with the convex hull of the zero set of f .
Now if Condition holds, t h e n it is obvious t h a t any nonvertical extreme ray of a convex set epi f** is contained in co(gr f ) . By Proposition 1, gr f contains all the extreme points of a convex set epi f**. This implies, in particular, t h a t every vertical e x t r e m e ray of epi f** is contained in epi f . Since epi f** does not contain lines, T h e o r e m 18.5 from [3] implies that conv(gr f U V) D epi f * * , a n d so conv(epi f U V) D gr f**. Here by V we denote t h e union of all vertical extreme rays of epi f * * .
Therefore, (x0, f**(x0)) E co(epi f ) whenever x0 E d o m f**. T h e n there exist points x l , . . . , x,, E IR n and n u m b e r s A1, . . . , Am > 0 a n d a l , . . . , a m > 0 such that ~ k = l Ak = 1 a n d
m
+
k----1
If some of the n u m b e r s a k , k = 1, . . . , m , are positive, then we have
m m
9 0 = Z and > Z
k----1 k----I
This contradicts the definition of the convexification of f**.
Now let us draw a hyperplane of s u p p o r t of the convex set at (x0, f * * ( x 0 ) ) . Using the Carath6odory theorem (see [3]), it can be easily shown t h a t the n u m b e r m in the representation (1) can be chosen so that r n < n - k l . []
1Editor's no~:e. T h e a u t h o r uses t h e s t a n d a r d n o t a t i o n of convex analysis ( c o n v , ri, etc., cf. [3]).
C o r o l l a r y 1. Suppose that f : R n --~ R is a Iower semicontinuous function linearly bounded below and such that the set d o m f C 1~ n contains no lines. Then the function f is overlapped i f and oniy K i t s graph contains an unbounded subset o f each nonvertical extreme r a y o f epi f * * .
P r o o f . It is easy to see that the set epi f** does not contain nonvertical lines. Indeed, otherwise the projection of this line on R n is a line contained in dom f . Obviously, epi f** contains no vertical lines also. T h e application of T h e o r e m 1 completes the proof. []
To formulate t h e next corollary we need some definitions.
A function f : N~_ ---* R is said to be sublinear if for any linear function l with nonnegative coefficients the difference f - I is b o u n d e d below by a constant. A function f is said to be strictly decreasing if f ( x ) < f ( y ) whenever x, y ~ ~ and y - x e R~ \ {0}.
C o r o l l a r y 2 (Shoples a n d Shubik [1, T h e o r e m 3]). If a function f : ~ --* R is continuous, sub//near, and strictly decreasing, then f is overlapped.
C o r o l l a r y 3. / / a lower semicontinuous function f : R n --. ~, satisfies the condition lim f ( x )
II ll - (3)
then f is overlapped.
P r o o f . It is cleax t h a t f is b o u n d e d below. It is easy to see t h a t t h e convexification of f** also satisfies condition (3). Thus, epi f** does not contain nonvertical extreme rays. T h e o r e m I implies that the function f is overlapped. []
Corollary 3 is stronger t h a n the above mentioned E k e l a n d - T e m a m l e m m a [2, p. 278]; in this l e m m a the following m o r e severe constraint is assumed instead of the growth condition (3):
lim f ( x ) - ce (a
>
I).II II
T h e following example shows that the growth condition is essential. E x a m p l e . Let
2Ix I for
Ixl
_< 1, f ( x ) = I x [ + 1 forIxl>
1. T h e n f**(x) = Ixl and f is not overlapped.T h e o r e m 2. Let f : R n ~ R be a function linearly bounded below with nontrivial space o f linearity o f the epigraph epi f**. Then f is overiapped i f and only i f the two following conditions hold: for each extreme r a y o f epi f** NL the projection PL" (gr fClepi f**) contains an u n b o u n d e d subset o f each extreme ray epi f** fl L -L a n d for any extreme point -2 of the set epi f** fl L -L the graph o f the function contains an enveloping subset o f -~ + L .
T h e proof of this t h e o r e m is based on T h e o r e m 1 and T h e o r e m 8.1 from [3].
References
1. L. S. Shoples and M. Shubik, Econometrica, 34, 805-827 (1966).2. I. Ekeland a n d R. T e m a m , Convex Analysis and Variational Problems (Studies in Mathematics and its Applications,
Vol. 1 N o r t h - H o l l a n d , A m s t e r d a m - O x f o r d , and American Elsevier P u b l . Co., New York (1976). 3. R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, N.J. (1970).
M. V. LOMONOSOV M o s c o w STATE UNIVERSITY
Translated by S. S. Anisov 521
