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Fibre tilings

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

Peter McMullen
Peter McMullen
Affiliation:
Department of Mathematics, University College London, Gower Street, London. WCIE 6BT, England.
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Abstract

Generalizing an earlier notion of secondary polytopes, Billera and Sturmfels introduced the important concept of fibre polytopes, and showed how they were related to certain kinds of subdivision induced by the projection of one polytope onto another. There are two obvious ways in which this concept can be extended: first, to possibly unbounded polyhedra, and second, by making the definition a categorical one. In the course of these investigations, it became clear that the whole subject fitted even more naturally into the context of finite tilings which admit strong duals. In turn, this new approach provides more unified and perspicuous explanations of many previously known but apparently quite disparate results.

MSC classification

Secondary: 52C22: Tilings in $n$ dimensions

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Type
Research Article
Information
Mathematika , Volume 50 , Issue 1-2 , December 2003 , pp. 1 - 33
Copyright
Copyright © University College London 2003

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References

1.Aurenhammer, F.. A criterion for the affine equivalence of cell complexes in Rd and convex polyhedra in Rd+1. Discrete Comput. Geom. 2 (1987), 49–64.CrossRefGoogle Scholar
2.Betke, U.. Mixed volumes of polytopes. Archiv Math. 58 (1992), 388–391.CrossRefGoogle Scholar
3.Billera, L. J., Gel'fand, I. M. and Sturmfels, B.. Duality and minors of secondary polyhedra. J. Combinat. Theory B 57 (1993), 258–268.CrossRefGoogle Scholar
4.Billera, L. J. and Sturmfels, B.. Fiber polytopes. Annals Math. 135 (1992), 527–549.CrossRefGoogle Scholar
5.Billera, L. J. and Sturmfels, B.. Iterated fiber polytopes. Mathemalika 41 (1994), 348–363.CrossRefGoogle Scholar
6.Gel'fand, I. M., Zelevinski, A. V. and Kapranov, M. M.. Newton polytopes of principal A-determinants. Soviet Math. Doklady 40 (1990), 287–281.Google Scholar
7.Goodey, P. R. and Weil, W.. Translative and kinematic formulae for support functions, II. Geom. Dedicata (to appear).Google Scholar
8.Grünbaum, B.. Convex Polytopes. Wiley-Interscience (London, 1967). (2nd edition, ed. Kaibel, V., Klee, V. L. and Ziegler, G. M., Springer, 2003.)Google Scholar
9.Hadwiger, H., Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer (Berlin, 1957).CrossRefGoogle Scholar
10.Huber, B. and Sturmfels, B., A polyhedral method for solving sparse polynomial systems. Math. Comp. 64 (1995). 1541–1555.CrossRefGoogle Scholar
11.McMullen, P., On zonotopes. Trans. Amer. Math. Soc. 159 (1971), 91–109.CrossRefGoogle Scholar
12.McMullen, P.. Representations of polytopes and polyhedral sets. Geom. Ded. 2 (1973), 83–99.CrossRefGoogle Scholar
13.McMullen, P., Transforms, diagrams and representations. In Contributions to Geometry (ed. Tölke, J. and Wills, J. M.), Birkhäuser (Basel, 1979), 92–130.CrossRefGoogle Scholar
14.McMullen, P.. The polytope algebra. Advances Math. 78 (1989), 76–130.CrossRefGoogle Scholar
15.McMullen, P., Valuations and dissections. Chapter 3.6 in Handbook of Convex geometry (ed. Gruber, P. M. and Wills, J. M.), North-Holland (1993), 933–988.CrossRefGoogle Scholar
16.McMullen, P., Duality, sections and projections of certain euclidean tilings. Geom. Dedicata 49 (1994), 183–202.CrossRefGoogle Scholar
17.McMullen, P.. Mixed fibre polytopes. Discrete Comput. Geom. 32 (2004), 521–532.CrossRefGoogle Scholar
18.McMullen, P. and Shephard, G. C., Diagrams for centrally symmetric polytopes. Mathematika 15 (1968). 123–138.CrossRefGoogle Scholar
19.McMullen, P. and Walkup, D. W., A generalized lower-bound conjecture for simplicial polytopes. Mathematika 18 (1971), 264–273.CrossRefGoogle Scholar
20.Rockafellar, R. T., Convex Analysis. Princeton University Press (1970).CrossRefGoogle Scholar
21.Rybnikov, K.. Stresses and liftings of cell-complexes. Discrete Comput. Geom. 21 (1999), 481–517.CrossRefGoogle Scholar
22.Schneider, R., Polytopes and Brunn-Minkowski theory. In Polytopes: Abstract, Convex and Computational (eds. Bisztriczky, T., McMullen, P., Schneider, R. and Weiss, A. Ivić), NATO ASI Series C 440, Kluwer (Dordrecht etc., 1994), 273–299.CrossRefGoogle Scholar
23.Schneider, R., Mixed polytopes. Discrete Comput. Geom. 29 (2003), 575–593.CrossRefGoogle Scholar
24.Shephard, G. C., Combinatorial properties of associated zonotopes. Canad. J. Math. 24 (1974), 302–321.CrossRefGoogle Scholar
25.Weil, W., Translative and kinematic integral formulae for support functions. Geom. Dedicata 57 (1995), 91–103.CrossRefGoogle Scholar
26.Ziegler, G. M., Lectures on Polytopes. Springer (New York, 1995).CrossRefGoogle Scholar