Journal of Combinatorial Theory, Series B 92 (2004) 1–12
The facets and the symmetries of the
and Samuel Fiorini
partement de Mathe
matique, cp. 216, Universite
Libre de Bruxelles, Boulevard du Triomphe,
1050 Brussels, Belgium
Department of Mathematics and Computer Science, HG 9.35, Technical University Eindhoven,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Received 7 November 2001
Available online 18 May 2004
Characterizing the size-independent model for approval voting of Falmagne and
Regenwetter (J. Math. Psychol. 40 (1996) 152) was shown by Doignon and Regenwetter
(J. Math. Psychol. 41 (1997) 171) to be equivalent to determining all facets of the approval-
voting polytope. Here, we prove that the facets of this polytope correspond in a natural way to
certain antichains in a power set. Several results on the approval-voting polytope are then
derived. For instance, all facet-deﬁning inequalities are characterized, and the groupof
automorphisms is completely described. On the other hand, providing an explicit listing of
all of the facets is shown to be at least as intricate as listing all connected graphs on a given,
r 2004 Elsevier Inc. All rights reserved.
MSC: 06A07; 52B12; 52B15
Keywords: Approval-voting polytope; Facet; Automorphism of convex polytope
ARTICLE IN PRESS
E-mail addresses: email@example.com (J.-P. Doignon), sﬁorini@ulb.ac.be (S. Fiorini).
Research done while visiting Eindhoven University of Technology as a postdoc of CWI in Amsterdam.
This postdoc position was ﬁnancially supported by the European Community via DONET (Contract ERB
TMRX-CT98-0202), a project in the Training and Mobility of Researchers Programme. Present address:
Libre de Bruxelles, De
partement de Mathe
matique, c.p. 216, Boulevard du Triomphe, 1050
0095-8956/$ - see front matter r 2004 Elsevier Inc. All rights reserved.