On complements of convex polyhedra as polynomial and regular images of ${\mathbb{R}}^n$

Abstract

In this work we prove constructively that the complement $\R^n\setminus\pol$ of a convex polyhedron $\pol\subset\R^n$ and the complement $\R^n\setminus\Int(\pol)$ of its interior are regular images of ${\mathbb{R}}^n$. If $\pol$ is moreover bounded, we can assure that $\R^n\setminus\pol$ and $\R^n\setminus\Int(\pol)$ are also polynomial images of ${\mathbb{R}}^n$. The construction of such regular and polynomial maps is done by double induction on the number of \em facets \em (faces of maximal dimension) and the dimension of $\pol$; the careful placing (\em first \em and \em second trimming positions\em) of the involved convex polyhedra which appear in each inductive step has interest by its own and it is the crucial part of our technique.

Cited by in corpus (3)

• Unbounded convex polyhedra as polynomial images of Euclidean spaces
• On polynomial images of a closed ball
• Surjective Nash maps between semialgebraic sets