Necessary and sufficient conditions for rigidity of the perimeter inequality under spherical symmetrisation are given. That is, a characterisation for the uniqueness (up to orthogonal transformations) of the extremals is provided. This is obtained through a careful analysis of the equality cases, and studying fine properties of the circular symmetrisation, which was firstly introduced by Pólya in 1950.

中文翻译：

---

球对称下周长不等式的刚度

给出了球对称下周边不等式刚度的充要条件。即，提供了对极值的唯一性（直到正交变换）的表征。这是通过对等式的仔细分析，并研究圆对称的优良性质而获得的，这是由Pólya在1950年首次提出的。