Journal of Global Optimization

, Volume 23, Issue 2, pp 111–137

Downward Sets and their separation and approximation properties

  • J.-E. Martínez-Legaz
  • A.M. Rubinov
  • I. Singer

DOI: 10.1023/A:1015583411806

Cite this article as:
Martínez-Legaz, JE., Rubinov, A. & Singer, I. Journal of Global Optimization (2002) 23: 111. doi:10.1023/A:1015583411806


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 ft (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.

Abstract convex function Abstract convex set Downward set Min-type coupling function Plus-Minkowski gauge 

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • J.-E. Martínez-Legaz
    • 1
  • A.M. Rubinov
    • 2
  • I. Singer
    • 3
  1. 1.CODE and Departament d'Economia i d'Història EconòmicaUniversitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.School of Information Technology and Mathematical SciencesUniversity of Ballarat, BallaratVictoriaAustralia
  3. 3.Institute of MathematicsBucharestRomania

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