# Analytic content and the isoperimetric inequality in higher dimensions

## TL;DR

This paper proves a conjecture linking the analytic content of smooth domains in higher dimensions to the isoperimetric inequality, using innovative methods from partial balayage and optimal transport theory.

## Contribution

It introduces a new proof connecting analytic content with geometric inequalities in higher dimensions through advanced mathematical techniques.

## Key findings

- Established the conjecture relating analytic content and isoperimetric inequality in higher dimensions.
- Developed a novel proof combining partial balayage and optimal transport theory.
- Enhanced understanding of geometric analysis in smooth domains.

## Abstract

This paper establishes a conjecture of Gustafsson and Khavinson, which relates the analytic content of a smoothly bounded domain in $\mathbb{R}^{N}$ to the classical isoperimetric inequality. The proof is based on a novel combination of partial balayage with optimal transport theory.