Positivity 11 (2007), 469–475
2007 Birkh¨auser Verlag Basel/Switzerland
Inequalities Characterizing Coisotone Cones
in Euclidean Spaces
Sandor Z. N´emeth
Abstract. The isotone projection cone, deﬁned by G. Isac and A. B. N´emeth,
is a closed pointed convex cone such that the order relation deﬁned by the
cone is preserved by the projection operator onto the cone. In this paper the
coisotone cone will be deﬁned as the polar of a generating isotone projection
cone. Several equivalent inequality conditions for the coisotonicity of a cone
in Euclidean spaces will be given.
Mathematics Subject Classiﬁcation (2000). 47H07, 47H99, 47H09, 46A40,
Keywords. Coisotone cones, Isotone projection cones, Latticial cones, Polar of
The metric projection onto a closed convex set in a Hilbert or Banach space has
been profoundly studied by several authors (see for example [1–3, 11–14, 16]).
A particularly important special case is the metric projection onto a closed
convex cone in a Hilbert space. This particular case is very important in the theory
of complementarity problems since, by a classical theorem of Moreau, every com-
plementarity problem is equivalent to a ﬁxed point problem expressed by using
the projection operator onto the cone deﬁning the complementarity problem. We
remark that several problems of mechanics, engineering and economics can be
modeled by using complementarity theory.
Although the special case of a projection onto a closed convex cone in a
Hilbert space has been extensively investigated by Zarantonello in , the con-
nection between the projection operator and the ordering deﬁned by a cone has
been ﬁrst considered by G. Isac and A. B. N´emeth in [4–8]. They raised the fol-
lowing very natural question: If K is a pointed closed convex cone and P
projection onto the cone, then for which cones K is the order “≤” deﬁned by K
Thanks are due to A. B. N´emeth who draw the author’s attention on the relation of latticial
cones generated by vectors with pairwise non-accute angles with the theory of isotone cones.