Aequationes Mathematicae 31 (1986) 322 University of Waterloo

0001-9054/86/003321-1 $1,50 + 0.20/0 © 1986 Birkhiiuser Verlag, Basel

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Midpoint convex functions majorized by midpoint concave functions

KAZIMIERZ NIKODEM

Let R denote the set of all real numbers and assume that D is a convex and open subset of R". A function f : D ~ R is said to be midpoint convex (midpoint concave)

1 f ( x ) + f ( y ) ] ( f ( ~ _ f f _ ) > ~ [ f ( x ) + f ( y ) ] ) , f o r a l l x , y~D;a func t i on i f f f ( ~ - ~ ) ~< ~[

a : R " ~ R is said to be additive iff a(x + y) = a(x)+ a(y), for all x ,y~R" . The following theorem holds.

THEOREM 1. I f a midpoint convex function f l :D---* ff~ is majorized on D by a

midpoint concave function f 2 : D ~ g~, then there exist an additive function a : R" ~ R, a convex function o a : D ~ R and a concave function o2 : D ~ R such that

f l (x ) = a(x) + O1(x) and f2(x) = a(x) + 02(x),for all x e D .

We say that a set-valued function F : D ~ 2 R is midpoint convex iff

F ( ~ - ) D 2 [ F ( x ) + F ( y ) ] , for all x, y e D . Denote by C ( R ) t h e family of all \ - - / - -

compact and non-empty subsets of R. Using Theorem 1 we obtain the following characterization of midpoint convex set-valued functions.

THEOREM 2. A set-valued function F: D --~ C(•) is midpoint convex i f and only

if it is of the form F(x) = a(x) + [Ox(X),O2(x)],xeD, where a: R " ~ R is an additive function, 91:D ~ R is a convex function and g2:D ~ I~ is a concave function.

Manuscript received March 5, 1985.

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