  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
$\mathbb{R}^n /\!/G$ 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. Br\"ocker 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 Br\"ocker concerning the
genericity of his result.
