# Rigidity for perimeter inequality under spherical symmetrisation

## TL;DR

This paper characterizes when the perimeter inequality remains rigid under spherical symmetrisation, providing conditions for the uniqueness of extremal sets through analysis of equality cases and properties of circular symmetrisation.

## Contribution

It offers necessary and sufficient conditions for rigidity in perimeter inequalities under spherical symmetrisation, advancing understanding of extremal set uniqueness.

## Key findings

- Conditions for rigidity are fully characterized.
- Analysis of equality cases reveals when extremals are unique.
- Provides insights into properties of circular symmetrisation.

## Abstract

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 P\'olya in 1950.