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Journal of Convex Analysis 19 (2012), No. 2, 485--496
Copyright Heldermann Verlag 2012

Lexicographical Representation of Convex Sets

Juan Enrique Martínez-Legaz
Departament d'Economia i d'Història Econòmica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

José Vicente-Pérez
Departamento de Estadística e Investigación Operativa, Universidad de Alicante, 03080 Alicante, Spain

We introduce two new families of properties on convex sets of Rn, in order to establish new theorems regarding open and closed separation of a convex set from any outside point by linear operators from Rn to Rm, in the sense of the lexicographical order of Rm, 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.

Keywords: Convex sets, open lexicographical separation, closed lexicographical separation.

MSC: 52A20, 90C25

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