Abstract
The main contribution of the paper is the proof that any element in the convex hull of a decomposably antichain-convex set is Pareto dominated by at least one element of that set. Building on this result, the paper demonstrates the disjointness of the convex hulls of two disjoint decomposably antichain-convex sets, under the assumption that one of the two sets is upward. These findings are used to obtain a number of consequences on: the structure of the set of Pareto optima of a decomposably antichain-convex set; the separation of two decomposably antichain-convex sets; the convexity of the set of maximals of an antichain-convex relation; the convexity of the set of maximizers of an antichain-quasiconcave function. Emphasis is placed on the invariance of the solution set of a problem under its “convexification.” Some entailments in the field of mathematical economics of the results of the paper are briefly discussed.
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Notes
A variant of that definition had already appeared in [14] to prove a fixpoint theorem.
This result has consequences of interest for economic theory. Note, for instance, that Theorem 3.2 allows to generalize Proposition 5.F.2 in [24] by replacing in its statement “convex” with “decomposably \(\mathrm {I\!R}_{+}^{n}\)-antichain-convex” (because any production vector that is profit-maximizing on \({\text {*}}{conv}(Y)\) must be profit-maximizing on Y).
For a concrete example of application of the separation result shown in Remark 4.3, see Theorem 7 in [12] and note that the mentioned separation result can replace part 2 of Theorem 4 in [12] in the proof of Theorem 7 in [12]: in such a proof, at least one of the sets \(\hat{A}\) and B has nonempty interior when V is not finite-dimensional.
Namely, \(h(a)\le h(b)\) for all \((a,b)\in U\times U\) such that \(a\le b\).
When \(V=\mathrm {I\!R}^{n}\), the value of the price functional at the consumption \(x\in X\) specifies the expenditure px given by the scalar product of \(p\in \mathrm {I\!R}^{n}\) and x: the vector p is called a price.
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The authors would like to thank the reviewers, an Associate Editor, and an Editor-in-Chief, for valuable suggestions which improved the paper.
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Communicated by Xiaoqi Yang.
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Ceparano, M.C., Quartieri, F. On Pareto Dominance in Decomposably Antichain-Convex Sets. J Optim Theory Appl 186, 68–85 (2020). https://doi.org/10.1007/s10957-020-01696-9
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DOI: https://doi.org/10.1007/s10957-020-01696-9