Convexity of individual consumer sets, and of the overall economy's production possibilities set plays is important role for proofs to obtain. Although the convexity requirement on the production possibilities set is only required to prove that any optimum can be supported by a market equilibrium under a certain price vector.

Only other comment is to point out that, according to Debreu, optimums are in general

*not comparable*to one another (except in trivial case where everyone is simply indifferent between the two optimums). Although, it is not entirely clear to me how this squares with the idea of Pareto dominance which is sometimes used to eliminate some market equilibria. To compare between two optimums would require, I think, being able to make inter-personal comparisons of utility...

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