Algebraic vertices of non-convex polyhedra

Akopyan, A. and Bárány, Imre and Robins, S. (2017) Algebraic vertices of non-convex polyhedra. ADVANCES IN MATHEMATICS, 308. pp. 627-644. ISSN 0001-8708

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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. © 2016 Elsevier Inc.

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