Journal of Convex Analysis 16 (2009), No. 3, 749--766
Copyright Heldermann Verlag 2009
When is a Convex Cone the Cone of all the Half-Lines Contained in a Convex Set?
Emil Ernst
Université de Aix-Marseille, UMR6632, 13397 Marseille, France
Emil.Ernst@univ-cezanne.fr
Michel Volle
Université d'Avignon et des Pays de Vaucluse, 74 Rue Louis Pasteur, 84029 Avignon, France
Michel.Volle@univ-avignon.fr
We prove that every convex cone V of a real vector space X possessing an uncountable Hamel basis may be expressed as the cone of all the half-lines contained within some convex subset C of X (in other words, V is the infinity cone to C). This property does not hold for lower-dimensional vector spaces; more precisely, a convex cone V in a vector space X with a denumerable basis is the infinity cone to some convex subset of X if and only if V is the union of a countable ascending sequence of linearly closed cones, while a convex cone V in a finite-dimensional vector space X is the infinity cone to some convex subset of X if and only if V is linearly closed.
Keywords: Infinity cone, recession analysis, spreading cover.
MSC: 26B99, 46N10, 49J99
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