Analytic content and the isoperimetric inequality in higher dimensions

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.

15 pages (There was a gap in a proof in the first version. The main result is unchanged, but the proof has been corrected in the later versions.)