Constructively describing orbit spaces of finite groups by few inequalities

Let $G$ be a finite group acting linearly on ${\mathbb{R}}^n$. A celebrated Theorem of Procesi and Schwarz gives an explicit description of the orbit space ${\mathbb{R}}^n\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$ as a basic closed semi-algebraic set. We give a new proof of this statement and another description as a basic closed semi-algebraic set using elementary tools from real algebraic geometry. Bröcker was able to show that the number of inequalities needed to describe the orbit space generically depends only on the group $G$. Here, we construct such inequalities explicitly for abelian groups and in the case where only one inequality is needed. Furthermore, we answer an open question raised by Bröcker concerning the genericity of his result.