A quantitative dimension free isoperimetric inequality for the fractional Gaussian perimeter

TL;DR

This paper establishes a dimension-free quantitative isoperimetric inequality for the Gaussian fractional perimeter, providing bounds that depend only on the Gaussian volume and fractional parameter, not on the dimension.

Contribution

It introduces a dimension-independent inequality for Gaussian fractional perimeter using extension techniques, advancing understanding of isoperimetric problems in Gaussian spaces.

Findings

Proves a quantitative isoperimetric inequality for Gaussian fractional perimeter.

The inequality's constant depends only on Gaussian volume and fractional parameter.

The exponent of the Fraenkel asymmetry is not sharp.

Abstract

We prove a quantitative isoperimetric inequality for the Gaussian fractional perimeter using extension techniques. Though the exponent of the Fraenkel asymmetry is not sharp, the constant appearing in the inequality does not depend on the dimension but only on the Gaussian volume of the set and on the fractional parameter.