, Volume 23, Issue 2, pp 111-137

Downward Sets and their separation and approximation properties

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We develop a theory of downward subsets of the space ℝI, where I is a finite index set. Downward sets arise as the set of all solutions of a system of inequalities x∈ℝI,ft(x)≤0 (t∈T), where T is an arbitrary index set and each f t (t∈T) is an increasing function defined on ℝI. These sets play an important role in some parts of mathematical economics and game theory. We examine some functions related to a downward set (the distance to this set and the plus-Minkowski gauge of this set, which we introduce here) and study lattices of closed downward sets and of corresponding distance functions. We discuss two kinds of duality for downward sets, based on multiplicative and additive min-type functions, respectively, and corresponding separation properties, and we give some characterizations of best approximations by downward sets. Some links between the multiplicative and additive cases are established.