Convex hull-like property and supported images of open sets

Abstract

In this note, as a particular case of a more general result, we obtain the following theorem: Let $\Omega\subseteq {\bf R}^n$ be a non-empty bounded open set and let $f:\overline {\Omega}\to {\bf R}^n$ be a continuous function which is $C^1$ in $\Omega$. Then, at least one of the following assertions holds: $(a)$ $f(\Omega)\subseteq \hbox {conv}(f(\partial \Omega))\ .$ $(b)$ There exists a non-empty open set $X\subseteq \Omega$, with $\overline {X}\subseteq \Omega$, satisfying the following property: for every continuous function $g:\Omega\to {\bf R}^n$ 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 $h:\Omega\to {\bf R}$ is a non-negative function, for each $u\in C^2(\Omega)\cap C^1(\overline {\Omega})$ satisfying in $\Omega$ the Monge-Amp\`ere equation $$u_{xx}u_{yy}-u_{xy}^2=h\ ,$$ one has $$\nabla u(\Omega)\subseteq \hbox {conv}(\nabla u(\partial\Omega))\ .$$