In this exposition, we investigate in an extended framework the problem of simultaneous best approximation. Refinement of a formula for the subdifferential of restricted radius of ap-bounded set leads us to explore some interesting results on strong uniqueness of simultaneous best approximants. In particular, we also. deal here with simultaneous best approximation of certain sets of vector-valued functions and establish strong uniqueness of order 2 for such approximants under suitable Haar-like hypotheses.