Abstarct: The range of ‎a‎ linear operator over a polyhedron, considered in this dissertation, is a convex subset H of a finite dimensional subspace S of the ambient inner product space X‎. ‎According to the reduction principle, the best approximation to a point of X from H equals the best approximation Ps(x) from H‎. From a geometric point of view different locations of Ps(x) with respect to H, these differences may have their bearings on the computation best approximation to a point of X from H of according to the reduction principle, ‎Using the Boyle-Dykstra theorem‎,by reducing the computation to a problem of finding best approximations x from H ‎. ‎The result is finally applied to real-life data from oil industry in that a solution is presented for a very important problem in oil‎ - ‎and gas production operations called the reconciliation problem‎, ‎where the contribution of individual wells to a measured total production has to be assessed‎.