A new, self-contained proof of Shahgholian's theorem using the thickness function

TL;DR

This paper offers a new, self-contained proof of Shahgholian's theorem on quadrature surfaces, utilizing the thickness function and level set methods to simplify the approach and enhance conceptual understanding.

Contribution

The authors develop a novel proof that avoids complex techniques like the moving plane method, providing a more intuitive and geometric perspective.

Findings

The proof confirms the geometric structure of quadrature surfaces under overdetermined conditions.

It demonstrates that level sets are parallel to the convex hull of the measure's support.

The approach simplifies understanding of Shahgholian's theorem through level set and maximum principle methods.

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.