The number of geometric bistellar neighbors of a triangulation
type: Item
instance of: scholarly article
title: The number of geometric bistellar neighbors of a triangulation
zbMATH Open document ID: 0935.52011
DOI: 10.1007/PL00009405
author: Jesús A. De Loera
author: J. Urrutia
author: Francisco Santos
published in: Discrete \& Computational Geometry
publication date: 1999-03-30T00:00:00Z
review text: A regular (or coherent) triangulation of a configuration of \(n\) points in \(\mathbb R^d\) arises from the projection of a simplicial polytope with \(n\) vertices in \(\mathbb R^{d+1}\). For straightforward geometric reasons, such a triangulation must have at least \(n-d-1\) bistellar neighbours, obtainable from it by bistellar operations. A non-regular triangulation, though, may have fewer bistellar neighbours. However, the authors show here that the result still holds if \(d=2\), or if \(d=3\) and the points are in convex position (lying on the boundary of their convex hull) with no three collinear. On the other hand, if \(d=3\) and even one point lies in the interior of the convex hull, then the result may fail. Lifting counterexamples to \(\mathbb R^4\) gives triangulations of simplicial 4-polytopes with \(n\) vertices and fewer than \(n-4\) bistellar neighbours.
Mathematics Subject Classification ID: 52B05
zbMATH DE Number: 1271013
zbMATH Keywords: polytope
zbMATH Keywords: triangulation
zbMATH Keywords: bistellar operation
zbMATH Keywords: bistellar neighbour
reviewed by: Peter McMullen
MaRDI profile type: Publication
full work available at URL: pl00009405
OpenAlex ID: W2021488116
Recommended article: Q4386081
Recommended article: ON THE NUMBER OF TRIANGULATION SIMPLEXES
Recommended article: The Number of Triangulations on Planar Point Sets
Recommended article: On numbers of pseudo-triangulations
Recommended article: On counting triangulations in \(d\) dimensions
Recommended article: Q4940023
Recommended article: Counting triangulations of planar point sets
Recommended article: Explicit enumeration of triangulations with multiple boundaries
Recommended article: Enumeration of bipartite non-crossing geometric graphs
description: scientific article; zbMATH DE number 1271013