The polytope of all triangulations of a point configuration
type: Item
instance of: scholarly article
title: The polytope of all triangulations of a point configuration
zbMATH Open document ID: 0852.52007
author: Serkan Hoşten
author: Bernd Sturmfels
author: Jesús A. De Loera
author: Francisco Santos
published in: Documenta Mathematica
publication date: 1996-07-07T00:00:00Z
full work available at URL: 04.html
review text: Let \(A = \{a_1, \ldots, a_n\}\) be a point configuration in \(\mathbb R^d\). The set of regular triangulations of \(A\) is well-understood by its bijection with the vertices of the so-called secondary polytope. Here the authors are mainly concerned with the set of non-regular triangulations of \(A\). More precisely, they investigate the convex hull \(P_A\) of the 0-1 incidence vectors of all triangulations of \(A\), being isomorphic to the so-called universal polytope which was introduced by \textit{L. J. Billera, P. Filliman} and \textit{B. Sturmfels} in [Adv. Math. 83, No. 2, 155--179 (1990; Zbl 0714.52004)]. E.g., the affine span of \(P_A\) is described in terms of the cocircuits of the oriented matroid of \(A\). Its intersection with the positive orthant is a quasi-integral polytope \(Q_A\) whose integral hull equals \(P_A\). The smallest example, where \(P_A\) and \(Q_A\) differ, is described, too. Furthermore, the known duality theory for regular triangulations is extended to all triangulations. Finally, some applications are discussed, e.g. to optimization problems referring to the set of all triangulations.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52B40
Mathematics Subject Classification ID: 90C27
zbMATH DE Number: 897859
zbMATH Keywords: secondary polytope
zbMATH Keywords: regular triangulation
zbMATH Keywords: chamber complex
zbMATH Keywords: incidence vector
zbMATH Keywords: point configuration in d-dim. Euclidean space
zbMATH Keywords: triangulations
zbMATH Keywords: universal polytope
zbMATH Keywords: oriented matroid
MaRDI profile type: Publication
Recommended article: Q4705029
Recommended article: Q5491019
Recommended article: Triangulations. Structures for algorithms and applications
Recommended article: Q4225317
Recommended article: Duality and minors of secondary polyhedra
Recommended article: Constructions and complexity of secondary polytopes
Recommended article: Q4401004
Recommended article: Q4952601
Recommended article: Triangulations of oriented matroids
Recommended article: Q3364072
description: scientific article; zbMATH DE number 897859