# On the refinements of a polyhedral subdivision.

Collectanea Mathematica (2001)

- Volume: 52, Issue: 3, page 231-256
- ISSN: 0010-0757

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topSantos, Francisco. "On the refinements of a polyhedral subdivision.." Collectanea Mathematica 52.3 (2001): 231-256. <http://eudml.org/doc/42771>.

@article{Santos2001,

abstract = {Let pi: P --> Q be an affine projection map between two polytopes P and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by pi (or pi-induced) and the fiber polytope of the projection: a polytope Sygma(P,pi) of dimension dim(P)-dim(Q) whose faces are in correspondence with the coherent pi-induced subdivisions (or pi-coherent subdivisions). In this paper we investigate the structure of the poset of pi-induced refinements of a pi-induced subdivision. In particular, we define the refinement polytope associated to any pi-induced subdivision S, which is a generalization of the fiber polytope and shares most of its properties. As applications to the theory we prove that if a point configuration has non-regular subdivisions then it has non-regular triangulations and we provide simple proofs of the existence of non-regular subdivisions for many particular point configurations.},

author = {Santos, Francisco},

journal = {Collectanea Mathematica},

keywords = {Geometría proyectiva; Politopos; Refinamiento; Poliedro; refinement polytope; fiber polytope; polyhedral subdivision; Baues conjecture},

language = {eng},

number = {3},

pages = {231-256},

title = {On the refinements of a polyhedral subdivision.},

url = {http://eudml.org/doc/42771},

volume = {52},

year = {2001},

}

TY - JOUR

AU - Santos, Francisco

TI - On the refinements of a polyhedral subdivision.

JO - Collectanea Mathematica

PY - 2001

VL - 52

IS - 3

SP - 231

EP - 256

AB - Let pi: P --> Q be an affine projection map between two polytopes P and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by pi (or pi-induced) and the fiber polytope of the projection: a polytope Sygma(P,pi) of dimension dim(P)-dim(Q) whose faces are in correspondence with the coherent pi-induced subdivisions (or pi-coherent subdivisions). In this paper we investigate the structure of the poset of pi-induced refinements of a pi-induced subdivision. In particular, we define the refinement polytope associated to any pi-induced subdivision S, which is a generalization of the fiber polytope and shares most of its properties. As applications to the theory we prove that if a point configuration has non-regular subdivisions then it has non-regular triangulations and we provide simple proofs of the existence of non-regular subdivisions for many particular point configurations.

LA - eng

KW - Geometría proyectiva; Politopos; Refinamiento; Poliedro; refinement polytope; fiber polytope; polyhedral subdivision; Baues conjecture

UR - http://eudml.org/doc/42771

ER -