ABSTRACT. Many interesting and important problems of best approximation are included in (or can be reduced to) one of the following type: in a Hilbert space X, find the best approximation P_{K}(x) to any x ∈ X from the set K := C ∩ A^{−1}(b), where C is a closed convex subset of X, A is a bounded linear operator from X into a finite-dimensional Hilbert space Y, and b ∈ Y. The main point of this paper is to show that P_{K}(x) is identical to P_{C}(x + A^{∗}y)—the best approximation to a certain perturbation x + A^{∗}y of x—from the convex set C or from a certain convex extremal subset C_{b} of C. The latter best approximation is generally much easier to compute than the former. Prior to this, the result had been known only in the case of a convex cone or for special data sets associated with a closed convex set. In fact, we give an intrinsic characterization of those pairs of sets C and A^{−1}(b) for which this can always be done. Finally, in many cases, the best approximation P_{C}(x+A^{∗}y) can be obtained numerically from existing algorithms or from modifications to existing algorithms. We give such an algorithm and prove its convergence.