Projection operators in the Weihrauch lattice

In this paper we study, for n >= 1, the projection operators over R^n, that is the multi-valued functions that associate to x ∈ R^n and A ⊆ R^n closed, the points of A which are closest to x. We also deal with approximate projections, where we content ourselves with points of A which are almost the closest to x. We use the tools of Weihrauch reducibility to classify these operators depending on the representation of A and the dimension n. It turns out that, depending on the representation of the closed sets and the dimension of the space, the projection and approximate projection operators characterize some of the most fundamental computational classes in the Weihrauch lattice.