On approximate solutions of systems of linear inequalities

Abstract

Let ~x ~ b be a. consis tent syst em of linp,ar inequal ities. The principal res ult is a quantitative formulatIOn of th e fa ct t hat if x "almost " sa t isfi es the inequ a lit ies t hen x is "close" t o a solu t ion. It is fur ther shown how it is possible in cert ain cases to e~timate the $ize of the vector joining x to the nearest solu t ion from t he magn itude of t he p osit ive coordi nates of A x -b.