Anisotropic best τC -approximation in normed spaces

The notions of the τ C -Kolmogorov condition, the τ C -sun and the τ C -regular point are introduced, and the relationships between them and the best τ C -approximation are explored. As a consequence, characterizations of best τ C -approximations from some kind of subsets (not necessarily convex) are obtained. As an application, a characterization result for a set C to be smooth is given in terms of the τ C -approximation.