Algebraic vertices of non-convex polyhedra Journal Article


Author(s): Akopyan, Arseniy; Bárány, Imre; Robins, Sinai
Article Title: Algebraic vertices of non-convex polyhedra
Affiliation IST Austria
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.
Keywords: Polytope algebra; Vertices; Tangent cones; Fourier–Laplace transform
Journal Title: Advances in Mathematics
Volume: 308
ISSN: 1090-2082
Publisher: Academic Press  
Date Published: 2017-02-21
Start Page: 627
End Page: 644
URL:
DOI: 10.1016/j.aim.2016.12.026
Notes: The first author was supported by People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007–2013) under REA grant agreement n°[291734]. The second author acknowledges support from ERC Advanced Research Grant no. 267165 (DISCONV) and by Hungarian National Research Grant K 111827. The third author was supported in part by ICERM, Brown University, and in part by the FAPESP grant Proc. 2103/03447-6, Brasil. A part of the paper was completed while the authors were supported by Singapore MOE Tier 2 Grant MOE2011-T2-1-090 (ARC 19/11) and the Institute for Mathematical Sciences at the National University of Singapore, under the program “Inverse Moment Problems: the Crossroads of Analysis, Algebra, Discrete Geometry and Combinatorics”.
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