On the convex hull of projective planes
RAIRO - Operations Research - Recherche Opérationnelle, Tome 42 (2008) no. 3, pp. 285-289.

We study the finite projective planes with linear programming models. We give a complete description of the convex hull of the finite projective planes of order 2. We give some integer linear programming models whose solution are, either a finite projective (or affine) plane of order n, or a (n+2)-arc.

DOI : 10.1051/ro:2008023
Classification : 5299, 0599
Mots clés : convex hull, finite projective plane
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Maurras, Jean-François; Nedev, Roumen. On the convex hull of projective planes. RAIRO - Operations Research - Recherche Opérationnelle, Tome 42 (2008) no. 3, pp. 285-289. doi : 10.1051/ro:2008023. http://www.numdam.org/articles/10.1051/ro:2008023/

[1] O. Anglada and J.F. Maurras, Enveloppe convexe des hyperplans d'un espace affine fini, avec Olivier Anglada. RAIRO-Oper. Res. 37 (2003) 213-219. | Numdam | MR | Zbl

[2] D. Avis, http://cgm.cs.mcgill.ca/ avis/C/lrs.html.

[3] D. Avis and K. Fukuda, A pivoting algorithm for convex hulls and vertex enumeration of arrankements and polyhedra. Discrete Comput. Geom. 8 (1992) 295-313. | MR | Zbl

[4] I. Bárány and Pór, 0-1 polytopes with many facets, Adv. Math. 161 (2001) 209-228. | MR | Zbl

[5] R.H. Bruck and H.J. Ryser, The nonexistence of certain finite projective planes. Can. J. Math. 1 (1949) 88-93. | MR | Zbl

[6] F.C. Bussemaker and J.J. Seidel, Symmetric Hadamard matrices of order 36. Report 70-WSK-02, TH Eindhoven, July (1970). | MR | Zbl

[7] T. Christof, www.zib.de/Optimization/Software/porta.

[8] K. Fukuda, http://cs.mcgill.ca/ fukuda/soft/cdd.

[9] P.B. Gibbons, Computing Techniques for the Construction and Analysis of Block Designs1976).

[10] T.R. Kirkman, On a problem in combinations. Camb. Dublin Math. J. 2 (1847) 191-204.

[11] C.W.H. Lam, The Search for a Finite Projective Plane of Order 10. Am. Math. Mon. 98 (1991) 305-318. | MR | Zbl

[12] M. Limbos, Projective embeddings of small Steiner triple systems. Ann. Discrete Math. 7 (1980) 151-173. | MR | Zbl

[13] R.A Mathon, K.T. Phelps and A. Rosa, Small Steiner triple systems and their properties. Ars Combinatoria 15 (1983) 3-110. | MR | Zbl

[14] J.F. Maurras, An exemple of dual polytopes in the unit hypercube. Ann. Discrete Math. 1 (1977) 391-392. | MR | Zbl

[15] J.F. Maurras, The Line Polytope of a finite Affine Plane. Discrete Math. 115 (1993) 283-286. | MR | Zbl

[16] T.S. Motzkin, H. Raiffa, G.L. Thompson and R.M. Thrall, The double description method, in H.W. Kuhn and A.W. Tucker, Eds., Contributions to theory of games, Vol. 2, Princeton University Press, Princeton (1953). | MR | Zbl

[17] H.S. White, F.N. Cole and L.D. Cummings, Complete classification of the triad systems on fifteen elements. Mem. Nat. Acad. Sci. U.S.A. 14, 2nd memoir (1919) 1-89.

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