Gale diagrams

2006 ◽  
pp. 37-45
Author(s):  
Rekha Thomas
Keyword(s):  
Gale Diagrams

Polytopes with an Axis Of Symmetry

1970 ◽  
Vol 22 (2) ◽  
pp. 265-287 ◽  
Author(s):  
P. McMullen ◽  
G. C. Shephard
Keyword(s):  
Symmetry Group ◽  
Linear Subspace ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Simple Manner ◽  
Gale Transform ◽  
Enumeration Problems ◽  
Axis Of Symmetry ◽  
Axes Of Symmetry ◽  
Gale Diagrams

During the last few years, Branko Grünbaum, Micha Perles, and others have made extensive use of Gale transforms and Gale diagrams in investigating the properties of convex polytopes. Up to the present, this technique has been applied almost entirely in connection with combinatorial and enumeration problems. In this paper we begin by showing that Gale transforms are also useful in investigating properties of an essentially metrical nature, namely the symmetries of a convex polytope. Our main result here (Theorem (10)) is that, in a manner that will be made precise later, the symmetry group of a polytope can be represented faithfully by the symmetry group of a Gale transform of its vertices. If a d-polytope P ⊂ Ed has an axis of symmetry A (that is, A is a linear subspace of Ed such that the reflection in A is a symmetry of P), then it is called axi-symmetric. Using Gale transforms we are able to determine, in a simple manner, the possible numbers and dimensions of axes of symmetry of axi-symmetric polytopes.

Geochemical Phase Diagrams and Gale Diagrams

2003 ◽  
Vol 64 (1) ◽  
pp. 231-259
Author(s):  
J. H. Stout ◽  
P. H. Edelman ◽  
S. W. Peterson ◽  
V. Reiner
Keyword(s):  
Phase Diagrams ◽  
Gale Diagrams

scholarly journals Strongly Robust Toric Ideals in Codimension 2

2019 ◽  
Vol 10 (1) ◽  
pp. 128-136 ◽  
Author(s):  
Seth Sullivant
Keyword(s):  
Positive Answer ◽  
Homogeneous Ideal ◽  
Toric Ideals ◽  
Grobner Basis ◽  
Toric Ideal ◽  
Generating Set ◽  
Graver Basis ◽  
Minimal Generating Set ◽  
Gale Diagrams ◽  
Universal Gröbner Basis

A homogeneous ideal is robust if its universal Gröbner basis is also a minimal generating set.  For toric ideals, one has the stronger definition: A toric ideal is strongly robust if its Graver basis equals the set of indispensable binomials.  We characterize the codimension 2  strongly robust toric ideals by their Gale diagrams.  This give a positive answer to a question of Petrovic, Thoma, and Vladoiu in the case of codimension 2 toric ideals.

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