BEST APPROXIMATION IN L(1)(I,X)

Let X be a Banach space and G a closed subspace of X. The subspace G is called proximinal in X if for every x is an element of X there exists at least one y is an element of G such that \\x - y\\ = d(x,G) = inf{\\x - z\\: z is an element of G}. It is an open problem whether L(1)(I,G) is proximinalin L(1)(I,X) if G is proximinal in X, where I is the unit interval with the Lebesgue measure. In this paper, we prove the proximinality of L(1)(I,G) in L(1)(I,X) for a class of proximinal subspaces G in X.