• Polyhedra
• Polynomial images
• Real algebraic geometry
• Regular images
• semialgebraic geometry

Modern real algebraic geometry was born with Tarski's article, where it is proved that the image of a semialgebraic set under a polynomial map is a semialgebraic set. We are interested in studying what might be the "inverse problem" to Tarski's result, that is to represent semialgebraic sets as polynomial or regular images of Euclidean spaces.
In 2002 Fernando and Gamboa answered both these questions. It constituted the starting point of the systematic study of the problem of representing semialgebraic sets as polynomial or regular images of Euclidean spaces.
Them, joinly with Ueno, have attempted to understand better polynomial and regular images of $\R^n$, in the last two decades.
In this dissertation we will focus on the regular case, and we will provide a complete computation, following the works of Fernando, Gamboa and Ueno; of the invariant $r$ for large families of semialgebraic sets with piecewise linear boundary: convex polyhedra, their interiors, their complements, the complement of their interiors.