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The slimmest arrangements of hyperplanes

I: Geometric lattices and projective arrangements

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Abstract

We show how the results of Dowling and Wilson on Whitney numbers in ‘The slimmest geometric lattices’ imply minimum values for the numbers of k-dimensional flats and d-dimensional cells of a projective d-arrangement of hyperplanes and for the number of d-cells missed by an extra hyperplane. Their theorems also characterize the extremal arrangements. We extend their lattice results to doubly indexed Whitney numbers; thence we obtain minima for the number of k-dimensional cells and the number of pairs of flats with x \(\subseteq\) y and dim x=k, dim y=l. The lower bounds are in terms of the rank and number of points of the geometric lattice, or the dimension d and the number of hyperplanes of the arrangement. The minima for k-cells were conjectured by Grünbaum; R. W. Shannon proved the minima for k-dimensional flats and cells and characterized attainment for the latter by a more strictly geometric, non-latticial technique.

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References

  1. Basterfield, J. G. and Kelly, L. M.: ‘A Characterization of Sets of n Points which Determine n Hyperplanes’, Proc. Cambridge Philos. Soc. 64 (1968), 585–588. (MR 38, # 2040.)

    Google Scholar 

  2. Birkhoff, G.: Lattice Theory (3rd edn). Amer. Math. Soc. Colloq. Publ. Vol. 25, American Mathematical Society, Providence, R.I., 1967. (MR 37, # 2638.)

    Google Scholar 

  3. Bland, R. G. and Las Vergnas, M.: ‘Orientability of Matroids’, J. Comb. Theory Ser. B 24 (1978), 94–123. (MR 58, # 5294.)

    Google Scholar 

  4. Brylawski, T.: ‘A Combinatorial Perspective on the Radon Convexity Theorem’, Geom. Dedicata 5 (1976), 459–466. (MR 55, # 13340.)

    Google Scholar 

  5. Cordovil, R.: ‘Sur l'évaluation t(M;2,0) du polynôme de Tutte d'un matroïde et une conjecture de B. Grünbaum relative aux arrangements de droites du plan’, Europ. J. Comb. 1 (1980), 317–322. (MR 82j: 05046.)

    Google Scholar 

  6. Crapo, H. H. and Rota, G. C.: Combinatorial Geometries, Preliminary edn, MIT Press, Cambridge, Mass., 1970. (MR 45, # 74.)

    Google Scholar 

  7. Dowling, T. A. and Wilson, R. M.: ‘The Slimmest Geometric Lattices’, Trans. Amer. Math. Soc. 196 (1974), 203–215. (MR 49, # 10579.)

    Google Scholar 

  8. Eckhoff, J.: ‘Radon's Theorem Revisited’, Contributions to Geometry, Proc. Geometry Symp., Siegen, 1978, J. Tölke and J. M. Wills, eds, Birkhäuser, Basel, 1979, pp. 164–185. (MR 81f: 52016.)

    Google Scholar 

  9. Folkman, J. and Lawrence, J.: ‘Oriented Matroids’, J. Comb. Theory Ser. B 25 (1978), 199–236. (MR 81g: 05045.)

    Google Scholar 

  10. Greene, C.: ‘A Rank Inequality for Finite Geometric Lattices’, J. Comb. Theory 9 (1970), 357–364. (MR 42, # 1727.)

    Google Scholar 

  11. Greene, C. and Zaslavsky, T.: ‘On the Interpretation of Whitney Numbers through Arrangements of Hyperplanes, Zonotopes, Non-Radon Partitions, and Orientations of Graphs’, Trans. Amer. Math. Soc. (in press).

  12. Grünbaum, B.: Convex Polytopes, Interscience, New York, 1967. (MR 37, # 2085.)

    Google Scholar 

  13. Grünbaum, B.: ‘Arrangements of Hyperplanes’, Proc. Second Louisiana Conf. on Combinatorics, Graph Theory and Computing (Baton Rouge, 1971), R. C. Mullin et al., eds, pp. 41–106. (MR 47, # 9428.)

  14. Las Vergnas, M.: ‘Matroïdes orientables’, C.R. Acad. Sci. Paris Ser. A 280 (1975), A61-A64. (MR 51, # 7910.)

    Google Scholar 

  15. Las Vergnas, M.: ‘Convexity in Oriented Matroids’, J. Comb. Theory Ser. B 29 (1980), 231–243. (MR 82f: 05027.)

    Google Scholar 

  16. Rota, G.-C.: ‘On the Foundations of Combinatorial Theory. I: Theory of Möbius Functions’, Z. Wahrschein. 2 (1964), 340–368. (MR 30, # 4688.)

    Google Scholar 

  17. Shannon, R. W.: ‘A Lower Bound on the Number of Cells in Arrangements of Hyperplanes’, J. Comb. Theory Ser. A 20 (1976), 327–335. (MR 53, # 14321.)

    Google Scholar 

  18. Welsh, D. J. A.: Matroid Theory, Academic, London, 1976. (MR 55, # 148.)

    Google Scholar 

  19. Zaslavsky, T.: ‘Facing up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes’, Mem. Amer. Math. Soc. 1(1975), Issue 1, No. 154. (MR 50, # 9603.)

  20. Zaslavsky, T.: ‘A Combinatorial Analysis of Topological Dissections’, Advances in Math. 25 (1977), 267–285. (MR 56, # 5310.)

    Google Scholar 

  21. Zaslavsky, T.: ‘Arrangements of Hyperplanes: Matroids and Graphs’, Proc. Tenth Southeastern Conf. on Combinatorics, Graph Theory and Computing (Boca Raton, 1979), F. Hoffman et al., eds, Utilitas Math. Publ. Inc., Winnipeg, Man., 1979, Vol. II, pp. 895–911. (MR 81h: 05043.)

    Google Scholar 

  22. Zaslavsky, T.: ‘The Slimmest Arrangements of Hyperplanes. II: Basepointed Geometric Lattices and Euclidean Arrangements’, Mathematika 28 (1981), 169–190.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, 231 West 18th Avenue, 43210, Columbus, Ohio, USA

    Thomas Zaslavsky

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  1. Thomas Zaslavsky
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Research supported by SGPNR Grant No. 74ZZ*08 and NSF Grant No. MCS-8003109.

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Zaslavsky, T. The slimmest arrangements of hyperplanes. Geom Dedicata 14, 243–259 (1983). https://doi.org/10.1007/BF00146905

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