A new, self-contained proof of Shahgholian's theorem using the thickness function
Analysis of PDEs
2026-04-01 v1
Abstract
This note presents a new, self-contained proof of Shahgholian's geometric theorem on quadrature surfaces using the thickness function and level set methods. By relying on a radial parametrisation and fundamental maximum principles, the proof avoids the technical complexity of the moving plane method. It provides a more conceptual view, revealing that the overdetermined condition forces all level sets to be parallel to the convex hull of the support of the measure.
Cite
@article{arxiv.2603.29084,
title = {A new, self-contained proof of Shahgholian's theorem using the thickness function},
author = {Mohammed Barkatou},
journal= {arXiv preprint arXiv:2603.29084},
year = {2026}
}