Blaschke-Santal\'o type inequalities and quermassintegral inequalities in space forms

TL;DR

This paper establishes new geometric inequalities for convex hypersurfaces in space forms, extending classical results to spherical, hyperbolic, and de Sitter geometries, with applications to Blaschke-Santaló and quermassintegral inequalities.

Contribution

It introduces novel identities and inequalities for convex hypersurfaces in various space forms, generalizing previous results and providing new geometric bounds.

Findings

Proved Blaschke-Santaló type inequalities in space forms

Established quermassintegral inequalities in hyperbolic/de Sitter space

Extended classical inequalities to spherical and hyperbolic geometries

Abstract

In this paper, we prove a family of identities for closed and strictly convex hypersurfaces in the sphere and hyperbolic/de Sitter space. As applications, we prove Blaschke-Santal\'o type inequalities in the sphere and hyperbolic/de Sitter space, which generalizes the previous work of Gao, Hug and Schneider \cite{GHS03}. We also prove the quermassintegral inequalities in hyperbolic/de Sitter space.