Algebraic vertices of non-convex polyhedra

TL;DR

This paper introduces the concept of algebraic vertices for generalized polyhedra, showing they are fundamental in defining the polyhedron and related indicator functions, with connections to Fourier--Laplace transforms.

Contribution

It defines algebraic vertices, proves their minimality in polyhedron representation, and links them to Fourier--Laplace transform properties.

Findings

Algebraic vertices form the minimal set to define a polyhedron.

Indicator functions of polytopes are linear combinations of simplices with algebraic vertices.

Fourier--Laplace transform characterizes algebraic vertices via tangent cones.

Abstract

In this article we define an algebraic vertex of a generalized polyhedron and show that it 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 $\mathbf{v}$ is an algebraic vertex of a generalized polyhedron $P$ if and only if the tangent cone of $P$, at $\mathbf{v}$, has non-zero Fourier--Laplace transform.