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.
@article{urn:eudml:doc:42771,
title = {On the refinements of a polyhedral subdivision.},
journal = {Collectanea Mathematica},
volume = {52},
year = {2001},
pages = {231-256},
zbl = {1012.52018},
mrnumber = {MR1885221},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:42771}
}
Santos, Francisco. On the refinements of a polyhedral subdivision.. Collectanea Mathematica, Tome 52 (2001) pp. 231-256. http://gdmltest.u-ga.fr/item/urn:eudml:doc:42771/