ABSTRACT. Let M be a (linear) subspace of the normed linear space X. The metric projection onto M is the set-valued mapping P_{M} : X → 2^{M} defined by

P_{M}(x) = {y ∈ M | ||x − y|| = inf_{m∈M} ||x − m||}.

M is called proximinal (resp. Chebyshev) if P_{M}(x) contains at least (resp. exactly) one point for each x in X. A linear selection for P_{M} is a linear map p : X → M such that p(x) ∈ P_{M}(x) for every x. It is easy to verify that every linear selection is a projection in the usual sense, i.e. p^{2} = p and p is continuous. (In fact, ||p|| ≤ 2). Thus linear selections are a special kind of continuous selection for P_{M}.

The purpose of this note is to provide a brief exposition of some of the results of [3] and to list some open problems. Full details along with related material can be found in [3].