Blaschke-Santaló type inequalities and quermassintegral inequalities in space forms

TL;DR

This work develops a unified duality framework for convex hypersurfaces in space forms, linking a hypersurface with its polar dual via the Gauss map and dual flows. It defines and analyzes quermassintegrals in spherical, hyperbolic, and de Sitter settings, establishing key invariants along dual curvature flows and deriving sharp Blaschke-Santaló type inequalities across sphere, hyperbolic, and de Sitter geometries. The authors obtain complete duality identities—$\tan\zeta_{n-k}(K)\tan\zeta_k(K^*)=1$ on the sphere and $\coth(\zeta_k(K))\tanh(\zeta_{n-k}(K^*))=1$ in hyperbolic/de Sitter—and translate these into a broad suite of quermassintegral inequalities, including full hyperbolic and de Sitter versions. The results extend classical Euclidean inequalities to curved space forms and provide a mechanism to transfer geometric inequalities between hyperbolic and de Sitter settings via duality, with explicit equality characterizations in terms of geodesic balls or coordinate slices.

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-Santaló 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.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1
• Theorem 1.4
• Theorem 1.5
• Remark 2
• Theorem 1.6
• Remark 3
• Conjecture 1.1