Boundary structure of convex sets in the hyperbolic space

TL;DR

This paper investigates the boundary properties of convex sets in hyperbolic space, focusing on curvature measures, normal points, and curvature characterizations, advancing the understanding of convex geometry in non-Euclidean settings.

Contribution

It introduces new results on curvature measure convergence, normal point analysis, and a generalized Gauss equation for convex surfaces in hyperbolic space.

Findings

Curvature measures converge under Hausdorff convergence of convex sets.

Characterization of normal points on convex boundaries.

A generalized Gauss equation for convex surfaces.

Abstract

We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian curvature measure.