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Iterated fiber polytopes

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  26 February 2010

Louis J. Billera  and
Bernd Sturmfels
Louis J. Billera
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853, U.S.A.
Bernd Sturmfels
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853, U.S.A.
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Abstract

The construction of the fiber polytope ∑(P, Q) of a projection π:P→Q of polytopes is extended to flags of projections. While the faces of the fiber polytope are related to subdivisions of Q induced by the faces of P, those of an iterated fiber polytope are related to discrete homotopies between polyhedral subdivisions. In particular, in the case of projections

starting with an (n + 1)-simplex, vertices of the successive iterates correspond to, respectively, subsets, permutations and sequences of permutations of an n-set. The first iterate will always be combinatorially an n-cube, and, under certain conditions, the second will have the structure of the (n−1)-dimensional permutohedron.

MSC classification

Secondary: 52B05: Combinatorial properties (number of faces, shortest paths, etc.)
Type
Research Article
Information
Mathematika , Volume 41 , Issue 2 , December 1994 , pp. 348 - 363
Copyright
Copyright © University College London 1994

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