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Gouveia, João; Macchia, Antonio; Wiebe, Amy

Combining realization space models of polytopes. (English) Zbl 1509.52013

Discrete Comput. Geom. 69, No. 2, 505-542 (2023).
Summary: 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 F. Criado and F. Santos [Exp. Math. 31, No. 2, 461–473 (2022; Zbl 1494.52007)], about the realizability of a family of prismatoids, in general in the negative by proving the non-realizability of one of them.

Cited in 1 Document

MSC:

52B99 Polytopes and polyhedra
52B11 \(n\)-dimensional polytopes
15B48 Positive matrices and their generalizations; cones of matrices
05E45 Combinatorial aspects of simplicial complexes
14P10 Semialgebraic sets and related spaces

Keywords:

polytopes; cones; realization spaces; Grassmannian; slack matrix; slack ideal; Gale transforms

Citations:

Zbl 1494.52007

Software:

SlackIdeals
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\textit{J. Gouveia} et al., Discrete Comput. Geom. 69, No. 2, 505--542 (2023; Zbl 1509.52013)
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References:

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