Convex hull-like property and supported images of open sets
Biagio Ricceri

TL;DR
This paper proves a convex hull-like property for continuous functions on bounded open sets in R^n, with applications to solutions of the Monge-Ampère equation in two dimensions, revealing geometric constraints on gradients.
Contribution
It establishes a new convex hull property for supported images of open sets and applies it to Monge-Ampère equations, extending geometric understanding of solutions.
Findings
Either the image of the open set is contained in the convex hull of its boundary image.
Or there exists a subset where the Jacobian determinant of certain functions vanishes for large parameters.
Applied to 2D Monge-Ampère equations, the gradient image is contained in the convex hull of boundary gradients.
Abstract
In this note, as a particular case of a more general result, we obtain the following theorem: Let be a non-empty bounded open set and let be a continuous function which is in . Then, at least one of the following assertions holds: There exists a non-empty open set , with , satisfying the following property: for every continuous function which is in , there exists such that, for each , the Jacobian determinant of the function vanishes at some point of . As a consequence, if and is a non-negative function, for each …
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