Systems of Inequalities Involving Convex

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 /i, ft, • • • , fm are said to be linearly independent, if no linear combination JZJlx Ai/,with real coefficients X,-, not all zero, can remain SiO throughout K. In the special case when K = X and when the//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 /,(x) = x' — l/(i+l) (l^ifkm) on the unit interval Ofkxfkl are linearly independent for any natural number m. In fact, if some m real numbers X< satisfy the relation YlT-i^x'-l/^+l)) ^0 for Ofkxfkl, then since we have JllYlT-i X,(x*'— l/(i + l))]dx = 0, the polynomial ^3TM i X,-(x' — l/(i + l)) must vanish identically and therefore all Xj = 0. The purpose of this note is to prove the following results concerning a system (1) of inequalities with convex functions /,defined on K.