Hoffman’s Least Error Bounds for Systems of Linear Inequalities

Abstract

Let E be a normed space, $a_1^* ,...,a_m^* \in E^* ,c_1 ,...,c_m \in R$ and $S = \left\{ {x \in E\left| {\left\langle {a_i^* ,x} \right\rangle - c_i \leqslant 0,1 \leqslant i \leqslant m} \right.} \right\} \ne \emptyset$. Let $\tau _* = \inf \left\{ {\tau \geqslant 0:dist\left( {x,S} \right) \leqslant \tau \max \left\{ {\left[ {\left\langle {a_i^* ,x} \right\rangle - c_i } \right]_ + :i = 1,...,m} \right\}\forall x \in E} \right\}$. We give some exact formulas for 7#x03C4;_✱.