An affine Orlicz Polya-Szego principle

TL;DR

This paper proves a generalized affine Orlicz Polya-Szego principle for all Orlicz Sobolev functions, extending previous results and deriving a new inequality for star bodies, thus advancing the understanding of geometric inequalities in analysis.

Contribution

It confirms the conjecture that the affine Orlicz Polya-Szego principle applies to all Orlicz Sobolev functions, unifying previous principles and deriving new geometric inequalities.

Findings

Established the affine Orlicz Polya-Szego principle for all Orlicz Sobolev functions.

Formulated a comprehensive affine Orlicz Polya-Szego principle encompassing previous cases.

Derived an Orlicz-Petty projection inequality for star bodies.

Abstract

The author established the affine Orlicz Polya-Szego principle for log-concave functions and conjectured that the principle can be extended to the general Orlicz Sobolev functions. In this paper, we confirm this conjecture completely. An affine Orlicz Polya-Szego principle, which includes all the previous affine Polya-Szego principles as special cases, is formulated and proved. As a consequence, an Orlicz-Petty projection inequality for star bodies is established.