Efficient sets are small
定义了Rp中的有效集，其中任何元素都不在所有维度上大于其他元素，并证明其Hausdorff维数至多为p-1，应用于多商品掠夺博弈中的稳定集。
We introduce efficient sets, a class of sets in Rp in which, in each set, no element is greater in all dimensions than any other. Neither differentiability nor continuity is required of such sets, which include: level sets of utility functions, quasi-indifference classes associated with a preference relation not given by a utility function, mean–variance frontiers, production possibility frontiers, and Pareto efficient sets. By Lebesgue’s density theorem, efficient sets have p-dimensional measure zero. As Lebesgue measure provides an imprecise description of small sets, we then prove the stronger result that each efficient set in Rp has Hausdorff dimension at most p−1. This may exceed its topological dimension, with the two notions becoming equivalent for smooth sets. We apply these results to stable sets in multi-good pillage games: for n agents and m goods, stable sets have dimension at most m(n−1)−1. This implies, and is much stronger than, the result that stable sets have m(n−1)-dimensional measure zero, as conjectured by Jordan.