The sharp quantitative barycentric isoperimetric inequality for bounded sets

TL;DR

This paper establishes the sharp quantitative barycentric isoperimetric inequality for bounded sets, extending previous 2D results to more general cases, providing precise bounds on how sets deviate from optimal shapes.

Contribution

It generalizes the sharp quantitative isoperimetric inequality involving barycentric asymmetry to higher dimensions for bounded sets.

Findings

Proves the sharp inequality for bounded sets in the barycentric case.

Extends previous 2D results to higher dimensions.

Provides explicit bounds on asymmetry and perimeter deviation.

Abstract

We prove the sharp quantitative isoperimetric inequality in the case of the barycentric asymmetry, for bounded sets. This generalizes the $2$-D case recently proved in~\cite{BCH}.