Fiber Polytopes for the Projections between Cyclic Polytopes

Christos A. Athanasiadis, Jesús A. De Loera, Victor Reiner, Francisco Santos

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9 Scopus citations

Abstract

The cyclic polytope C (n, d) is the convex hull of any n points on the moment curve {(t, t2, . . . , td) : t ∈ ℝ} in ℝd. For d′ > d, we consider the fiber polytope (in the sense of Billera and Sturmfels [6]) associated to the natural projection of cyclic polytopes π : C(n, d′) → C(n, d) which 'forgets' the last d′ - d coordinates. It is known that this fiber polytope has face lattice indexed by the coherent polytopal subdivisions of C(n, d) which are induced by the map π. Our main result characterizes the triples (n, d, d′) for which the fiber polytope is canonical in either of the following two senses: • all polytopal subdivisions induced by π are coherent, • the structure of the fiber polytope does not depend upon the choice of points on the moment curve. We also discuss a new instance with a positive answer to the generalized Baues problem, namely that of a projection π : P → Q where Q has only regular subdivisions and P has two more vertices than its dimension.

Original languageEnglish (US)
Pages (from-to)19-47
Number of pages29
JournalEuropean Journal of Combinatorics
Volume21
Issue number1
DOIs
StatePublished - Jan 2000

Bibliographical note

Funding Information:
The first author (CAA) was supported by a Mathematical Sciences Research Institute postdoctoral fellowship and a University of Pennsylvania Hans Rademacher Instructorship. Research of the second (JAL) and fourth (FA) authors was partially supported by the Geometry Center (NSF grant DMS-8920161). The third author (VR) was supported by a University of Minnesota McKnight-Land Grant Fellowship and Sloan Foundation Fellowship. The fourth author (FS) was supported by CajaCantabria and by grant PB97-0358 of Spanish Dirección General de Enseñanza Superior e Investigación Científica

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