On Well Posed Generalized Best Approximation Problems

Let C be a closed bounded convex subset of X with 0 being an interior point of C and p"C be the Minkowski functional with respect to C. Let G be a nonempty closed, boundedly relatively weakly compact subset of a Banach space X. For a point [email protected]?X, we say the minimization problem min"C(x, G) is well posed if there exists a unique point z such that p"C(z-x)[email protected]"C(x, G) and every sequence {z"n}@?G satisfying lim"n"->"~p"C(z"n-x)[email protected]"C(x, G) converges strongly to the point z, where @l"C(x, G)=inf"z"@?"Gp"C(z-x). Under the assumption that C is both strictly convex and Kadec, we prove that the set X"o(G) of all [email protected]?X such that the problem min"C(x, G) is well posed is a residual subset of X extending the results in the case that the modulus of convexity of C is strictly positive due to Blasi and Myjak. In addition, we also prove these conditions are necessary.