Author(s):

Akopyan, Arseniy; Bárány, Imre; Robins, Sinai

Article Title: 
Algebraic vertices of nonconvex 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 linecones.
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 nonzero Fourier–Laplace transform.

Keywords: 
Polytope algebra; Vertices; Tangent cones; Fourier–Laplace transform

Journal Title:

Advances in Mathematics

Volume: 
308

ISSN:

00018708

Publisher:

Elsevier

Date Published:

20170221

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. 2103/034476, Brasil.
A part of the paper was completed while the authors were supported by Singapore MOE Tier 2 Grant MOE2011T21090 (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”.

Open access: 
yes (repository) 