Precise estimates for certain distances in $\mathbb{R}^d$

TL;DR

This paper derives precise estimates for various intrinsic distances in convex and strongly convex domains in Euclidean space, including the Kobayashi-Hilbert, minimal, and $k$-quasi hyperbolic metrics, with applications in convex geometry.

Contribution

It provides sharp boundary estimates for these metrics and characterizes the $k$-quasi hyperbolic metric within convex geometry, advancing understanding of intrinsic distances.

Findings

Sharp boundary estimates for the Kobayashi-Hilbert metric near strongly convex points

Precise estimates for the minimal metric near convex and strongly minimally convex points

Characterization of the $k$-quasi hyperbolic metric in convex domains

Abstract

We provide sharp estimates for the intrinsic distances of Finsler metrics with precise boundary estimates. These metrics include the Kobayashi-Hilbert metric near strongly convex points, the minimal metric near convex and strongly minimally convex points, and the $k$-quasi hyperbolic metric in $k$-strongly convex domains. Finally, we prove a characterization result in convex geometry for the $k$-quasi hyperbolic metric.