Approximative properties of sets and continuous selections

Sets admitting a continuous selection of the operators of best and near-best approximation are studied. Michael's classical continuous selection theorem is extended to the case of a lower semicontinuous metric projection in finite-dimensional spaces (with no a priori convexity conditions on its values). Sufficient conditions on the metric projection implying the solarity of the corresponding set are put forward in finite-dimensional polyhedral spaces. Available results for suns V are employed to establish the existence of continuous selections of the relative (with respect to V) Chebyshev near-centre map and of the sets of relative (with respect to V)