Small sets in best approximation theory

The best approximation problem to a nonempty closed set in a locally uniformly convex Banach space is considered. The main result states that the set of points which have best approximation but the approximation problem is not well-posed is very small in a sense that it is σ-cone supported in the underlying space. This gives an improvement of an original result of Stečkin about the set of points with more than one best approximation which involves Baire categories. Examples on the necessity of some of the imposed conditions are provided.