# Algebraic vertices of non-convex polyhedra

Akopyan A, Bárány I, Robins S. 2017. Algebraic vertices of non-convex polyhedra. Advances in Mathematics. 308, 627–644.

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https://arxiv.org/abs/1508.07594
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*Journal Article*|

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*English*

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Author

Akopyan, Arseniy

^{ISTA}^{}; Bárány, Imre; Robins, SinaiDepartment

Abstract

In this article we define an algebraic vertex of a generalized polyhedron and show that the set of algebraic vertices is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope P is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of P. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier–Laplace transform. We show that a point v is an algebraic vertex of a generalized polyhedron P if and only if the tangent cone of P, at v, has non-zero Fourier–Laplace transform.

Publishing Year

Date Published

2017-02-21

Journal Title

Advances in Mathematics

Volume

308

Page

627 - 644

ISSN

IST-REx-ID

### Cite this

Akopyan A, Bárány I, Robins S. Algebraic vertices of non-convex polyhedra.

*Advances in Mathematics*. 2017;308:627-644. doi:10.1016/j.aim.2016.12.026Akopyan, A., Bárány, I., & Robins, S. (2017). Algebraic vertices of non-convex polyhedra.

*Advances in Mathematics*. Academic Press. https://doi.org/10.1016/j.aim.2016.12.026Akopyan, Arseniy, Imre Bárány, and Sinai Robins. “Algebraic Vertices of Non-Convex Polyhedra.”

*Advances in Mathematics*. Academic Press, 2017. https://doi.org/10.1016/j.aim.2016.12.026.A. Akopyan, I. Bárány, and S. Robins, “Algebraic vertices of non-convex polyhedra,”

*Advances in Mathematics*, vol. 308. Academic Press, pp. 627–644, 2017.Akopyan, Arseniy, et al. “Algebraic Vertices of Non-Convex Polyhedra.”

*Advances in Mathematics*, vol. 308, Academic Press, 2017, pp. 627–44, doi:10.1016/j.aim.2016.12.026.**All files available under the following license(s):**

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