I would like to solve the following (apparently not so simple) problem.
Given the following sets:
$\mathcal{X} = \{ x \in R^n | Ax \leq b \}$
$\mathcal{Y} = \{ y \in R^n | C y \leq d \}$
I would like to find the largest subset $\tilde{\mathcal{X}} \subseteq \mathcal{X} $ such that the set $\tilde{\mathcal{Y}}$, defined as the codomain of the smooth function
$ F(x) : \tilde{\mathcal{X}} \rightarrow \tilde{\mathcal{Y}}$
is a subset of $\mathcal{Y}$, i.e. $\tilde{\mathcal{Y}} \subseteq \mathcal{Y}$.
When I say largest subset I mean the subset with the maximum area.
NOTE: It is also fine parametrize the $\tilde{\mathcal{X}}$ such that $\tilde{\mathcal{X}}(s) = \{ x \in R^n , s \in R^p \; | \; Ax -b \leq s \; ; \; s \leq 0 \}$. In this case the problem would be to find the values of $s$ that maximize the area of $\tilde{\mathcal{X}}(s)$ and such that $\tilde{\mathcal{Y}}(s) \subseteq \mathcal{Y}$.