Continuous selections for metric projections in spaces of continuous functions and a disjoint leaves condition

Let T be a locally compact Hausdorff space and let G denote a finite-dimensional subspace of the space C0(T) of those real-valued continuous functions on T which vanish at infinity, and let the space be equipped with the uniform norm. Li [Continuous Selections for Metric Projections and Interpolating Subspaces, vol. 1, Approximation and Optimization, Verlag Peter Lang, Frankfurt, 1991] characterized those G with the property that there exists a continuous selection for the set valued metric projection PG:C0(T)→P(G) of C0(T) onto G. Using work of Fischer [Continuous selections for semi-infinite optimisation, in: Parametric Optimisation and Related Topics, Akademic-Verlag, Berlin, 1987, pp. 95–112] an alternative characterization is obtained. A direct proof that the two characterizing conditions are equivalent provides an alternative proof, shorter than Li's, of Li's characterization.