# The sharp quantitative isocapacitary inequality

## TL;DR

This paper establishes a precise quantitative relationship between the capacity difference of a set and a ball of equal volume and the set's asymmetry, confirming a longstanding conjecture.

## Contribution

It proves a sharp quantitative form of the isocapacitary inequality, linking capacity difference to Fraenkel asymmetry, and resolves a conjecture from 1991.

## Key findings

- Capacity difference bounds the square of Fraenkel asymmetry.
- Provides a positive answer to a 1991 conjecture.
- Sharp inequality with optimal constants.

## Abstract

We prove a sharp quantitative form of the classical isocapacitary inequality. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the square of the Fraenkel asymmetry of the set. This provides a positive answer to a conjecture of Hall, Hayman, and Weitsman (J. d' Analyse Math. '91).