An extension of the Sard–Smale Theorem to convex domains with an empty interior

A stumbling block in the modeling of competitive markets with commodity and price spaces of infinite dimensions, arises from having positive cones with an empty interior. This issue precludes the use of tools of differential analysis, ranging from the definition of a derivative, to the use of more sophisticated results needed to understand determinacy of equilibria and, more generally, the structure of the equilibrium set. To overcome these issues, this note extends the Preimage Theorem and the Sard–Smale Theorem to maps between convex sets that may have an empty interior.