Abstract

We introduce two new families of properties on convex sets of Rn\mathbb{R}^n, in order to establish new theorems regarding open and closed separation of a convex set from any outside point by linear operators from Rn\mathbb{R}^n to Rm\mathbb{R}^m, in the sense of the lexicographical order of Rm\mathbb{R}^m, for each m = 1, 2,..., n. We thus obtain lexicographical extensions of well known separation theorems for convex sets as well as characterizations of the solution sets of lexicographical (weak and strict) inequality systems defined by matrices of a given rank.

Author information

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Juan Enrique Martínez-Legaz

Departament d'Economia i d'Història Econòmica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
and: MOVE (Markets, Organizations and Votes in Economics)

JuanEnrique.Martinez.Legaz@uab.cat

José Vicente-Pérez

Departamento de Estadística e Investigación Operativa, Universidad de Alicante, 03080 Alicante, Spain

Jose.Vicente@ua.es

Keywords

  • Convex sets
  • open lexicographical separation
  • closed lexicographical separation

Mathematics Subject Classification

52A20, 90C25

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Suggested citation

J. E. Martínez-Legaz, J. Vicente-Pérez. “Lexicographical Representation of Convex Sets.” Journal of Convex Analysis 19 (2012), No. 2, 485–496. https://doi.org/10.68381/jca19026

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.