Systems of inequalities involving convex functions

Abstract

is said to be consistent, if an element xEK satisfying all m inequalities (i.e. a solution of (1)) does exist. Otherwise (1) is said to be inconsistent. A system (1) is said to be irreducibly inconsistent, if it is inconsistent and if every proper subsystem of (1) is consistent. The convex functions fi, f2, , fm are said to be linearly independent, if no linear combination ZJ-,1 Xif with real coefficients Xi, not all zero, can remain 20 throughout K. In the special case when K =X and when the fi's are linear forms on X, this definition of linear independence agrees with the usual one. Indeed, a linear form can remain >0 throughout the entire vector space X only when it is identically zero. Incidentally, we observe that there exist arbitrarily many linearly independent convex functions even on a one-dimensional convex set. For example, the m convex functions f,(x) =xi-1/(i+1) (1_i_m) on the unit interval 0?x?1 are linearly independent for any natural number m. In fact, if some m real numbers Xi satisfy the reIation ET1 Xi(xi-1/(i+1)) 2O0 for 0?x?<1, then since we have f['[E!1 1Xi(xi-1/(i+1))]dx=0, the polynomial E-1 X,(xi-1/(i+1)) must vanish identically and therefore all X, =0. The purpose of this note is to prove the following results concerning a system (1) of inequalities with convex functions f, defined on K.