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The Polytope of m-Subspacesof a Finite Affine Space

Published online by Cambridge University Press:  21 August 2007

Julie Christophe  and
Jean-Paul Doignon
Julie Christophe
Affiliation:
Université Libre de Bruxelles, c.p. 216, Bd du Triomphe, 1050 Bruxelles, Belgium; doignon@ulb.ac.be
Jean-Paul Doignon
Affiliation:
Université Libre de Bruxelles, c.p. 216, Bd du Triomphe, 1050 Bruxelles, Belgium; doignon@ulb.ac.be
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Abstract

The m-subspace polytope is defined as the convex hull of the characteristic vectors of all m-dimensional subspaces ofa finite affine space. The particular case of the hyperplane polytopehas been investigated by Maurras (1993) and Anglada and Maurras (2003), who gave a complete characterization of the facets. The general m-subspace polytope that we consider shows a much more involved structure, notably as regards facets. Nevertheless, several families of facets are established here. Then the group of automorphisms of the m-subspace polytope is completely described and the adjacency of vertices is fully characterized.

Keywords

Convex polytopefinite affine spacem-subspace polytope
Type
Research Article
Information
RAIRO - Operations Research , Volume 41 , Issue 3: Journées Polyèdres et Optimisation Combinatoire , July 2007 , pp. 317 - 344
Copyright
© EDP Sciences, ROADEF, SMAI, 2007

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References

O. Anglada, Quelques polyèdres combinatoires bien décrits. Ph.D. thesis, Université de la Méditerranée Aix-Marseille II, Faculté des Sciences de Luminy (2004).
Anglada, O. and Maurras, J.F., Enveloppe convexe des hyperplans d'un espace affine fini. RAIRO Oper. Res. 37 (2003) 213–219. CrossRef
E. Artin, Geometric Algebra. Wiley Classics Library. John Wiley & Sons Inc., New York (1988).
A. Beutelspacher and U. Rosenbaum, Projective Geometry: from foundations to applications. Cambridge University Press, Cambridge (1998).
Bolotashvili, G., Kovalev, M., and Girlich, E., New facets of the linear ordering polytope. SIAM J. Discrete Math. 12 (1999) 326–336. CrossRef
A. Brøndsted, An Introduction to Convex Polytopes. Graduate Texts in Mathematics, vol. 90. Springer-Verlag, New York (1983).
T. Christof, PORTA – a POlyhedron Representation Transformation Algorithm. version 1.3.2 (1999), written by T. Christof, maintained by A. Loebel and M. Stoer, available at http://www.informatik.uni-heidelberg.de/groups/comopt/software/PORTA/.
J. Christophe, Le polytope des sous-espaces d'un espace affin fini. Ph.D. thesis, Université Libre de Bruxelles, 2006. Accessible on line at http://theses.ulb.ac.be:8000/ETD-db/collection/available/ULBetd-01222007-165129/ro.html
Christophe, J., Doignon, J.-P. and Fiorini, S., The biorder polytope. Order 21 (2004) 61–82. CrossRef
Doignon, J.-P and Regenwetter, M., An approval-voting polytope for linear orders. J. Math. Psych. 41 (1997) 171–188. CrossRef
E. Gawrilow and M. Joswig, POLYMAKE: a software package for analyzing convex polytopes. Version 1.4 (Dec. 2000), available at url http://www.math-tu-berlin.de/diskregeom/polymake/.
B. Grünbaum, Convex Polytopes, Graduate Texts in Mathematics, vol. 221. Springer-Verlag, New York, second edition (2003).
Maurras, J.F., The line-polytope of a finite affine plane. Discrete Math. 115 (1993) 283–286. CrossRef
J.H. van Lint and R.M. Wilson, A Course in Combinatorics. Cambridge University Press, Cambridge, UK (1992).
G.M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152. Springer-Verlag (1995).