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

TL;DR

This paper demonstrates that the complements of convex polyhedra and their interiors in al^n are regular images, and polynomial images if bounded, constructed via a double induction method involving geometric positioning techniques.

Contribution

It provides a constructive proof that complements of convex polyhedra are regular or polynomial images of al^n, introducing a novel double induction approach with geometric positioning.

Findings

Complements of convex polyhedra are regular images of al^n.

Bounded polyhedra have polynomial images as complements.

The construction uses a double induction on facets and dimension.

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 $\R^n$. If $\pol$ is moreover bounded, we can assure that $\R^n\setminus\pol$ and $\R^n\setminus\Int(\pol)$ are also polynomial images of $\R^n$. 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.