Convex hull-like property and supported images of open sets

In this note, as a particular case of a more general result, we obtain the following theorem: Let $\mathrm{\Omega }\subseteq {\mathbf{R}}^n$ be a non-empty bounded open set and let $f:\bar{\mathrm{\Omega }}\to {\mathbf{R}}^n$ be a continuous function which is $C^1$ in $\mathrm{\Omega }$. Then, at least one of the following assertions holds: $(a)$ $f(\mathrm{\Omega })\subseteq \text{conv}(f(\mathrm{\partial }\mathrm{\Omega }))\mathrm{.}$ $(b)$ There exists a non-empty open set $X\subseteq \mathrm{\Omega }$, with $\overline {X}\subseteq \Omega$, satisfying the following property: for every continuous function $g:\mathrm{\Omega }\to {\mathbf{R}}^n$ which is $C^1$ in $X$, there exists $\tilde{\lambda }>0$ such that, for each $\lambda >\tilde{\lambda }$, the Jacobian determinant of the function $g+\lambda f$ vanishes at some point of $X$. As a consequence, if $n=2$ and $h:\mathrm{\Omega }\to \mathbf{R}$ is a non-negative function, for each $u\in C^2(\mathrm{\Omega })\cap C^1(\bar{\mathrm{\Omega }})$ satisfying in $\mathrm{\Omega }$ the Monge-Amp`ere equation $$u_{xx}{u}_{yy}-u_{xy}^2=h,$$ one has $$\mathrm{\nabla }u(\mathrm{\Omega })\subseteq \text{conv}(\mathrm{\nabla }u(\mathrm{\partial }\mathrm{\Omega }))\mathrm{.}$$