Best Approximation Problems in Compactly Uniformly Rotund Spaces

We investigate under what geometric conditions the best approximation problem to a nonempty closed subset of a real Banach space is generalized well-posed, or, more generally, the problem either has no solution or is generalized well-posed, for the majority of the points in the space. "Majority" is understood as a set whose complement in the space is σ-porous or σ-cone supported. Analogously to the case when uniqueness of the best approximation is considered, it turns out that certain local uniform, or uniform, properties of the norm of the underlying space have to be required.