On Complements of Convex Polyhedra as Polynomial and Regular Images of Rn

Abstract

In this work we prove constructively that the complement Rn \ K of a convex polyhedron K ⊂ Rn and the complement Rn \ Int(K) of its interior are regular images of Rn. If K is moreover bounded, we can assure that Rn \ K and Rn \ Int(K) are also polynomial images of Rn. The construction of such regular and polynomial maps is done by double induction on the number of facets (faces of maximal dimension) and the dimension of K; the careful placing (first and second trimming positions) 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.