Petty projection inequality on the sphere and on the hyperbolic space

TL;DR

This paper introduces projection bodies and Steiner symmetrization in spherical and hyperbolic spaces, proving new projection inequalities using geometric models like gnomonic projection and Poincaré model.

Contribution

It defines spherical and hyperbolic projection bodies and symmetrizations, establishing the Petty projection inequalities in these non-Euclidean geometries.

Findings

Established the spherical projection inequality.

Proved the hyperbolic projection inequality.

Extended Petty projection inequality to spherical and hyperbolic spaces.

Abstract

Using gnomonic projection and Poincar\'e model, we first define the spherical projection body and hyperbolic projection body in spherical space $\mathbb{S}^n$ and hyperbolic space $\mathbb{H}^n$, then define the spherical Steiner symmetrization and hyperbolic Steiner symmetrization, finally prove the spherical projection inequality and hyperbolic projection inequality.