We first characterize the region of the n-dimensional Euclidean space for which two optimization problems with the square distance function as common objective function, but different constraints, are equivalent. The affine hull of a certain face of a closed convex set \(C\subseteq {\mathbb {R}}^n\) is the constraint associated to one problem and the whole closed convex set C is the constraint associated to the other problem. Such optimization problems are best approximation problems which can be reformulated in terms of the metric projection. Using the language of the metric projection, we characterize the region of \({\mathbb {R}}^n\) which is metrically projected over a face of C in the same way that it is projected over the affine hull of the face itself. The metric projection over such a face is the one associated to the entire closed convex set, and the metric projection over the affine hull of such a face is the one associated to the affine hull. It turns out that this region is the closure of the inverse image, through the metric projection over the entire closed convex set, of the relative interior of the face. We also characterize analytically the closure of the regions of \({\mathbb {R}}^n\) that are projected over the relative interiors of the faces of a polyhedral set, through the metric projection of the polyhedral set itself. We show that these regions are polyhedral convex sets by explicitly characterizing them through systems of linear inequalities.
