Computing the distance to the functions with range in a given set in Lebesgue spaces of vector-valued functions

In this paper we give a formula for the distance from an element f of the Banach space L_p(\mu,X), 1 ≤ p ≤ \infty---where X is a Banach space and (T, M, \mu) is a positive measure space---to the subset L_p(\mu,S) of all functions whose range is (essentially) contained in a given nonempty subset S of X. This formula is in terms of the norm in L_p(\mu) of the distance function to S that is induced by f, namely, of the scalar-valued function d_f^S which maps t \in T into the distance from f(t) to S. Indeed, the sets S for which L_p(\mu,S) is nonempty are characterized. Furthermore, for such sets the function d_f^S is proved to be in L_p(\mu), and the distance from f to L_p(\mu,S) is proved to coincide with the norm of d_f^S in L_p(\mu). This generalizes previously obtained formulae---provided in [LC], 2.10 and in [L], Theorem 5---in which the distance to L_p(\mu,S) is computed in the special case of a vector subspace S of X.