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 subset of R(n) and a point p is an element of R(n) \ K, we construct a semialgebraic partition {A, B, T} of the boundary partial derivative 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 J 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).