Convex composite functions in Banach spaces and the primal lower-nice property

Primal lower-nice functions defined on Hilbert spaces provide examples of functions that are ``integrable'' (i.e. of functions that are determined up to an additive constant by their subgradients). The class of primal lower-nice functions contains all convex and lower-$C^2$ functions. In finite dimensions the class of primal lower-nice functions also contains the composition of a convex function with a $C^2$ mapping under a constraint qualification. In Banach spaces certain convex composite functions were known to be primal lower-nice (e.g. a convex function had to be continuous relative to its domain). In this paper we weaken the assumptions and provide new examples of convex composite functions defined on a Banach space with the primal lower-nice property. One consequence of our results is the identification of new examples of integrable functions on Hilbert spaces.