Please use this identifier to cite or link to this item: https://hdl.handle.net/10316/113917
Title: Combining Realization Space Models of Polytopes
Authors: Gouveia, João 
Macchia, Antonio
Wiebe, Amy
Keywords: Polytopes; Cones; Realization spaces; Grassmannian; Slack matrix; Slack ideal; Gale transforms
Issue Date: 2022
Publisher: Springer Nature
metadata.degois.publication.title: Discrete and Computational Geometry
metadata.degois.publication.volume: 69
metadata.degois.publication.issue: 2
Abstract: In this paper we examine four different models for the realization space of a polytope: the classical model, the Grassmannian model, the Gale transform model, and the slack variety. Respectively, they identify realizations of the polytopes with the matrix whose columns are the coordinates of their vertices, the column space of said matrix, their Gale transforms, and their slack matrices. Each model has been used to study realizations of polytopes. In this paper we establish very explicitly the maps that allow us to move between models, study their precise relationships, and combine the strengths of different viewpoints. As an illustration, we combine the compact nature of the Grassmannian model with the slack variety to obtain a reduced slack model that allows us to perform slack ideal calculations that were previously out of computational reach. These calculations allow us to answer the question of Criado and Santos (Exp. Math. (2019). https://doi.org/10.1080/10586458.2019.1641766), about the realizability of a family of prismatoids, in general in the negative by proving the non-realizability of one of them.
URI: https://hdl.handle.net/10316/113917
ISSN: 0179-5376
1432-0444
DOI: 10.1007/s00454-022-00379-8
Rights: openAccess
Appears in Collections:I&D CMUC - Artigos em Revistas Internacionais
FCTUC Matemática - Artigos em Revistas Internacionais

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