Curvatures of spheres in Hilbert geometry

TL;DR

This paper investigates the asymptotic behavior of various curvatures of hyperspheres and circles in Hilbert geometry, showing they tend to 1 as the radii grow large.

Contribution

It establishes the limiting behavior of normal, Rund, and Finsler curvatures for large-radius spheres in Hilbert geometry, a new insight into their geometric properties.

Findings

Normal curvatures tend to 1 for large radii

Rund curvature approaches 1 as radius increases

Finsler curvature also tends to 1 for large circles

Abstract

We prove that the normal curvatures of hyperspheres, the Rund curvature, and the Finsler curvature of circles in Hilbert geometry tend to 1 as the radii tend to infinity