The 1951 Arrow-Debreu treatment relies on the fact that Pareto optimal allocations lead to two sets being disjoint. If these sets are convex,⁶⁵ one appeals to the geometric insight that they can be separated by a line.⁶⁶ Thus, convexity assumptions have to be made on the basic data of the economy, namely on preferences and on technologies, and they ensure that the functional-analytic argument can be sustained. The content of the theorem can now be given an alternative form. Corresponding to every Pareto optimal allocation, there exists a system of prices—the separating line so to speak—such that decentralized self-interested decisions of consumers and producers lead to that allocation being sustained. Since such decisions imply that every agent equate his or her marginal rates to this price, the previous results discussed by Samuelson-Graaff seem to be contained in this reformulation. The rates, if they exist, can be equalized but they are no longer the issue; the fact that desirable outcomes can be sustained as individual maximizing behavior is the crucial insight. Convexity takes the center stage.

Khan - “The Irony in/of Economic Theory,” p. 780

⁶⁵: A set is convex if the line joining any two points chosen from the set also lies in the set. Thus a crescent is not a convex set whereas a disc is. See Rockafeller (1970) for details.

⁶⁶: This is nothing more profound than saying that one set lies on one side of the line and the other set lies on the other side of the line; see Rockafellar (1970) for details. Of course, with sufficient generality, the statement constitutes the geometric version of the fundamental Hahn-Banach theorem of functional analysis.