The geometry of linear infeasibility

摘要

The best approximation of an unsolvable system of linear equations is shown to lie in a set that is bounded by finite many hyperplanes but need not be convex. This candidate set, defined to be the polyhedral interior of the linear system, is the same for the best approximations with respect to all p-norms, 1⩽p⩽∞. Polyhedral considerations allow the treatment of several issues including the removal of linear equations to render the remaining system feasible.