On Perturbations in Systems of Linear Inequalities

We consider what happens to sets defined by systems of linear inequalities when elements of the system are perturbed. If $S = \{ x|Gx \leqq g,Dx = d\} $, and if ${S'}$ and ${S''}$ are defined in the obvious manner by perturbed matrices $G',G'',g',g'',D',D'',d',d''$, we show that, under certain hypotheses, to each element ${x'}$ in ${S'}$ there corresponds ${x''}$ in ${S''}$ with $\| {x' - x''} \| \leqq c\{ \| {G' - G''} \| + \| {g' - g''} \| + \| {D' - D''} \| + \| {d' - d''} \|\} (1 + \| {x'} \|)$ for some constant c depending on S.