Positivity 7: 61–72, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Finite Coverings by Cones and an Application in
STEF TIJS and HANS REIJNIERSE
Department of Econometrics and CentER, Tilburg University, P.O. Box 90153, 5000 LE Tilburg,
The Netherlands. E-mail: S.H.Tijs@uvt.nl; J.H.Reijnierse@uvt.nl
Abstract. This paper considers analogues of statements concerning compactness and ﬁnite cover-
ings, in which the roles of spheres are replaced by cones. Furthermore, one of the ﬁnite covering
results provides an application in Multi-Objective Programming; inﬁnite sets of alternatives are
reduced to ﬁnite sets.
JEL code: C00, C79
Key words: cones, ε-domination, ﬁnite covering, multi-objective programming
An elementary observation gives that each bounded set in a ﬁnite-dimensional real
space has a ﬁnite subset such that each element is close to one of the elements of
the subset, irrespective of the quantifying of the notion close.
If the boundedness is relaxed to upper boundedness, the statement can be re-
paired by replacing ‘close to’ by ‘almost dominated by’. This result is called the
ε-domination Theorem and has been ﬁrstly proved by Tijs . It has been used
to derive ε-equilibrium point theorems for two person noncooperative games and
resulted in a literature on approximate equilibria and other approximate solutions
Section 2 provides a new proof of the ε-domination Theorem. Furthermore, it
shows that the characterizing property of compactness that every open covering of
a compact set contains a ﬁnite subcovering, cannot be converted in a similar way.
After a presentation of some of these results at the converence ‘Variational
Methods and Optimization, in memory of Francesco Ferro (1946–1999)’ (Genova,
2000), J.-P. Penot phrased the question whether the ε-domination Theorem, which
deals with sets of the form a + R
can be generalized to ﬁnite coverings by sets
of the form a + K,inwhichK is a cone. First of all, the original set need not be
upper bounded, but K-bounded, i.e., covered by some positive variety a + K.
To our surprise, the answer turns out to be yes and no. Section 3 provides
a generalization concerning polyhedral cones. However, Section 4 shows that in