Optimal reconstruction of a Banach function space from a cone of nonnegative functions

We study the problem of constructing a minimal Banach function space containing a given cone of nonnegative measurable functions. For the associate function norm of the norm of an optimal space, we obtain general formulas and specify them in the case of a cone defined by an integral representation. We also consider the similar problem of constructing an optimal rearrangement invariant space and compare the descriptions obtained.