On convex polyhedra as regular images of ℝ ⁿ

Abstract

We show that convex polyhedra in R n and their interiors are images of regular maps R n → R n . As a main ingredient in the proof, given an n-dimensional, bounded, convex polyhedron K ⊂ R n and a point p ∈ R n \ K, we construct a semialgebraic partition {A, B, T} of the boundary ∂K of K determined by p, and compatible with the interiors of the faces of K, such that A and B are semialgebraically homeomorphic to an (n − 1)-dimensional open ball and T is semialgebraically homeomorphic to an (n − 2)-dimensional sphere. Finally, we also prove that closed balls in R n and their interiors are images of regular maps R n → R n .

On regulous and regular images of Euclidean spaces

“…• To prove (constructively) that large families of semialgebraic sets with piecewise linear boundary (convex polyhedra, their interiors, their complements and the interiors of their complements) are either polynomial or regular images of Euclidean spaces [FGU1,FGU4,FU1,FU2,FU5,U1,U2].…”

On the Set of Points at Infinity of a Polynomial Image of $${\mathbb R}^n$$ R n

“…Even though, we have approached the representation as polynomial or regular images of ample families of n-dimensional semialgebraic sets whose boundaries are piecewise linear. We have focused on: convex polyhedra, their interiors, their exteriors and the closure of their exteriors [FGU1,FU1,FU2,U2]. The proofs are constructive but the arguments are developed ad hoc.…”

On the complements of 3-dimensional convex polyhedra as polynomial images of ℝ3

“…Ample families. By showing constructively that ample families of significant semialgebraic sets are either polynomial or regular images of R n (see [Fe,FG1,FGU,FU2,U2]).…”