in fact, the orbit space $V/K$ and its principal stratum $\widehat \Sigma$ can be identified, respectively, with the semialgebraic sets $\lambda(V)$ and $\lambda(\Sigma)$:

[$${\widehat\Sigma \,=\,
\lambda(\Sigma)\,=\,\{\lambda\in\real^l\mid \widehat
\Lambda(\lambda)>0\},\qquad V/K= \overline{\widehat
\Sigma}\,=\,\lambda(V)\,=\,\{\lambda\in\real^l\mid
\widehat\Lambda(\lambda)\ge 0\},}\label{20}$$]{} the second set being the closure of the first one. Moreover, the definition of $\widehat{\mathcal V}$ and equation [(\[ins\])]{} imply

[$${\widehat{\mathcal V}=\lambda({\mathcal
V})\subseteq\lambda(\Sigma)\,=\,\widehat\Sigma.}\label{inshat}$$]{}

If $K$ is not coregular, only a local one-to-one parametrization can be obtained for $\widehat \Sigma$, by eliminating redundant elements in the set of parameters $(\lambda_1,\dots ,\lambda_l)$, through the solution of the algebraic relation(s) among the elements of the MIB $\{\lambda\}$, and imposing convenient semi-positivity and rank conditions on the matrix $\widehat\Lambda(\lambda)$.

A one-to-one local or global parametrization of $\widehat \Sigma$ yields, obviously, also a local or global one-to-one parametrization of $\widehat {\mathcal V}$, provided that additional restrictions are imposed on the range of $\lambda$, whenever $\widehat {\mathcal V}$ is a proper subset of $\widehat\Sigma$. In this case, the correct bounds can be obtained in the following way (we shall only consider the case of a coregular $K$, the extension of the results to non coregular $K$’s is straightforward, but, as just stressed, may lead to a loss of globality).

When $x$ spans $V$, 