The authors consider problems of characterization and uniqueness for best \(L^ 1\)-approximation to continuous functions, where the approximating sets are specified convex subsets of finite-dimensional subspaces. After a very interesting introduction in this area, they consider the problem of bounded coefficient approximation and best approximation under interpolatory constraints. In the following two sections are discussed the problem of best restricted range approximation. Section 5 contains the more theoretical aspects of this problem, while in Section 6 are embodied various examples. Finally, is showed how these results can be generalized to restricted range and derivative approximation.