Weak Chebyshev subspaces and continuous selections for the metric projection

Let G be an n-dimensional subspace of C [ a , b ] C[a,b] . It is shown that there exists a continuous selection for the metric projection if for each f in C [ a , b ] C[a,b] there exists exactly one alternation element g f {g_f} , i.e., a best approximation for f such that for some a ⩽ x 0 > ⋯ > x n ⩽ b a \leqslant {x_0} > \cdots > {x_n} \leqslant b , \[ ε ( − 1 ) i ( f − g f ) ( x i ) = ‖ f − g f ‖ , i = 0 , … , n , ε = ± 1. \varepsilon {( - 1)^i}(f - {g_f})({x_i}) = \left \| {f - {g_f}} \right \|,\quad i = 0, \ldots ,n,\varepsilon = \pm 1. \] Further it is shown that this condition is fulfilled if and only if G is a weak Chebyshev subspace with the property that each g in G, g ≠ 0 g \ne 0 , has at most n distinct zeros. These results generalize in a certain sense results of Lazar, Morris and Wulbert for n = 1 n = 1 and Brown for n = 5 n = 5 .