The 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 $\Omega \subseteq {\mathbf{R}}^n$ be a nonempty bounded open set, and let $f:\overline{\Omega }\to {\mathbf{R}}^n$ be a continuous function which is $C^1$ in $\Omega$. Then, at least one of the following assertions holds:

(a) $f(\Omega )\subseteq conv\left(f\right(\partial \Omega \left)\right)$.

(b) There exists a nonempty open set $X\subseteq \Omega$, with $\overline{X}\subseteq \Omega$, satisfying the following property: for every continuous function $g:\Omega \to {\mathbf{R}}^n$ which is $C^1$ in $X$, there exists $\tilde{\lambda }\ge 0$ 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 $h:\Omega \to \mathbf{R}$ is a nonnegative function, for each $u\in C^2(\Omega )\cap C^1\left(\overline{\Omega }\right)$ satisfying in $\Omega$ the Monge–Ampère equation

$$u_{xx}{u}_{yy}-u_{xy}^2=h,$$ one has

$$\nabla u(\Omega )\subseteq conv(\nabla u(\partial \Omega \left)\right).$$