The Petty projection inequality for sets of finite perimeter

TL;DR

This paper proves the Petty projection inequality for sets of finite perimeter using Steiner symmetrization, providing a new proof that avoids the affine Sobolev inequality and the functional Minkowski problem.

Contribution

It introduces a novel proof of the Petty projection inequality for finite perimeter sets based solely on Steiner symmetrization, bypassing traditional methods.

Findings

Established the Petty projection inequality for finite perimeter sets.

Demonstrated the inequality with respect to Steiner symmetrization.

Provided a new proof technique avoiding complex inequalities.

Abstract

The Petty projection inequality for sets of finite perimeter is proved. Our approach is based on Steiner symmetrization. Neither the affine Sobolev inequality nor the functional Minkowski problem is used in our proof. Moreover, for sets of finite perimeter, we prove the Petty projection inequality with respect to Steiner symmetrization.